Keywords: Controllability theory, network controllability, target controllability, driver nodes, biological networks, molecular networks, brain networks, 23, 24, 34-36, 41, 45, 47, 48, 50-58, 61-63, 81, 87-91 . ix, 54, 57, 58 DTI diffusion tensor imaging. x, 49, 50, 60, 83 fMRI functional magnetic resonance imaging. 2, 49, 50, 60, 61 invariant. iii, 2, 6, 7, 15 63 MDS minimum driver node set. 9, xi, 3, 30, 31, 33, 36-39

The possibility of using mathematical tools to describe and influence complex interconnected systems is getting more and more attainable. Methods based on network controllability to identify the nodes able to impact the state of a whole system are nowadays increasingly studied. However, the problem has a high combinatorial and numerical complexity because of the huge number of a priori equivalent solutions.

There has recently been a growing interest in finding the minimum number of inputs to control the whole or a part of the system, and in evaluating the ability of a single node in steering this process. However, specific problems have drawn less attention.

In some biological settings it may be required to act on a single node, and it may be of interest to affect only a well-defined subset of the units, a target set. This leads to a single input target control problem, where we can exploit biological constraints to study the relative importance of different driver nodes. This dissertation aims to apply controllability theory to biological networks in an original way, to understand what insight mathematical controllability theory can bring to biological networks, and to study the importance of different driver nodes in controlling a target set. We develop a heuristic that we call step-wise target controllability, which measures the centrality of a driver node as the number of targets it can control and provides a controllable configuration of targets. We show that this method is efficient for sparse directed networks. To test the theory in two empirical settings, we apply it to macrophage activation and to the aging of the brain. We find that this framework can identify dysregulated pathways in a molecular network of macrophage activation, and that it can characterize normal aging in the brain. Our method represents a practical answer to use to our advantage the complexity of the control problem, exploiting existing biological knowledge.

List of Figures

3.1

The intuition behind the controllability property of a system refers to the fact that (a) any initial state can be steered to any final state by choosing the input appropriately and (b) applying an external input to a node (here X 1 ), it is possible to constrain the state of the whole network. . [START_REF] Biggs | Algebraic Graph Theory[END_REF] 4.1 Methodological validation of step-wise target controllability. We start from a simple directed full binary tree, and we add a cycle among the first three nodes. The driver is the root of the tree, while we put a target node in each level of the tree. Target nodes are then ranked according to their height h. This configuration is fully target controllable by construction regardless of the tree's height h. However, for h = 5 (i.e. N = 63), the standard procedure computing the rank of the full Kalman controllability matrix cannot retrieve all the targets, as the rank computation is deficient due to numerical errors. Instead, by using the step-wise target controllability we can correctly identify the controllable targets up to h = 10, i.e. N = lustrates a network with one driver and a target set T = {t 1 , t 2 , t 3 } of cardinality S = 3. The Kalman condition informs us that only two targets are controllable from the driver, i.e. τ = rank(Q T ) = 2. However, there might be up to 3 equivalent configurations that are controllable, i.e. {t 1 , t 2 }, {t 1 , t 3 }, and {t 2 , t 3 }. For larger networks, the number of Kalman tests to perform can be prohibitive, i.e. ( S τ ). Panel b). By introducing a hierarchy among the target nodes, our step-wise method identifies the configuration with the most important nodes by performing only S tests (see Methods).

In this example, the first step considers the subgraph containing all the walks from the driver to the target set T = {t 1 }. The associated controllability matrix has full rank, i.e. rank(Q T ) = 1. The first target is therefore retained and the algorithm moves to Step 2, by constructing a new subgraph containing the walks from the driver to the target set T = {t 1 , t 2 }.

The rank of the new controllability matrix is now deficient and t 2 is not retained. In Step 3, the new subgraph contains the walks from the driver to T = {t 1 , t 3 }. Because rank(Q T ) is full and there are no more targets, the algorithm stops and returns the controllable configuration t 1 , t 3 . . . . . . com repository [START_REF] Raza | A logic-based diagram of signalling pathways central to macrophage activation[END_REF][START_REF] Robert | Macrophages.com: An on-line community resource for innate immunity research[END_REF]. The network consists of N = 101 nodes corresponding to genes involved in inflammation; for the sake of interpretation, they are organized in four classes, depending on their function in the cell.

Sensing genes are in the membrane of the cell and start a signaling pathway inside the cell, to the transcription factors, which promote the production of secreted molecules. There are L = 211 directed edges representing either activation or inhibition interactions between molecules (Methods).

The size of the nodes is proportional to their total degree k. Panel b)

shows gene activation computed as the ratio in expression between the "proinflammatory" and "alert" states, based on our RNA sequencing data, generated from monocyte-derived macrophages from blood samples of multiple sclerosis patients (n = 8) and healthy controls (n = 8) (Methods). Solid reports the coactive driver-target pairs, computed as significant Spearman correlations (p< 0.05) between the gene activation of controllable drivertarget pairs, for the healthy control (HC) (blue squares) and the multiple sclerosis (MS) (red squares) groups. White squares indicate that there is a controllable walk from the driver to the target, but that their correlation is not significant. Grey squares mean that there is no controllable walk for driver-target pairs. The size of the circles for driver nodes codes for their . 39 5.7 Step-wise target control centrality (τ ) averaged across systems and subjects, for the limbic and sensorimotor target sets. Panel a)

shows τ centrality values for each brain region when the limbic system is the target set. This case is also exemplary of the case in which one among the frontal, temporal, parietal, or occipital system is the target set. In these configurations, the sensorimotor system has the largest values of centrality, suggesting that it may be well-suited to control the other systems. On the other hand, for all target configurations the limbic system presents low centrality values. Panel b) shows τ centrality values for each brain region when the sensorimotor system is the target set. In this case the values are extremely low for all structural systems, suggesting that the sensorimotor system may be difficult to control. . . . . . . . . . . . . . . . . . . . . 51 5.8 Self-regulation score averaged across systems and subjects, at the varying of the target set. Self-regulation for a system is computed as the ratio between average τ in that system and total average τ . Thus, selfregulation above (below) 1 indicates that nodes in that system can control more (less) targets than an average node. Across all configurations apart for the limbic system, the target system itself has the largest values of selfregulation, partially explained by the fact that each target node can at least control itself. In all configurations, the limbic system has low self-regulation. and age, corrected for outdegree. The correction is aimed at disentangling the contribution of outgoing connections and that of the centrality.

The outdegree is relevant to the controllability property, in fact outgoing edges make possible to control other nodes. Panel a) shows brain regions that present a significant correlation (p < 0.05) for at least one target configuration. Bigger and darker dots correspond to regions of interest (ROIs) that are consistently significant across multiple target sets. Cool (warm) colors stand for negative (positive) correlations, white identifies ROIs for which the trend changes for different target sets. The ROI with a thick black border is the left middle temporal temporooccipital (LMTGtp) region. This region is located in the temporal system, and it is known to be involved in the default mode network. It is the region presenting the strongest negative correlation among the ones consistent for all six target systems. Panel b) shows the scatter plot of τ and age for LMTGtp when the target is the occipital system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.10 Absolute difference in step-wise target control centrality (τ ) between the simulation of a lesion in one hemisphere of the brain and the normal case. When simulating stroke, the hemisphere most impacted is the one with the lesion. The sensorimotor system appears to be the most affected, while the limbic system is the least affected. Attacking the sensorimotor system led to minor global losses, while attacking any of the other systems led to considerable global losses. Panels a-l) show how the absolute difference in centrality values is distributed on the connectomes at the varying of the target set, when the lesion is on the right (panels a-f), or on the left (panels g-l). Panels m-x) show averaged values of the absolute difference for the left (solid orange circles) and right (dotted blue crosses) hemispheres for the corresponding target set configurations, when the lesion is on the right (panels m-r), or on the left (panels s-x). Error bars stand for the standard error across subjects. . . . . . . . . . . . . . . . . . . . . . 56 SA1 Hierarchy among target genes. Genes corresponding to the 13 secreted molecules are ranked according to the absolute value of fold change ∆ in the gene activation between the multiple sclerosis (MS) group and the healthy control (HC) group (Methods). . . . . . . . . . . . . . . . . . . . . . 67 lable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table. . . . . . . . . . . SA7 Spearman cross-correlation p-values for multiple sclerosis (MS) patients for controllable driver-target node pairs. Values for pairs that were not controllable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table. . . . . . . . . . . SA8 Log-transformed gene expression for healthy controls (HC), in the 'alert'

Chapter 1

Introduction

For thousands of years, humans have modified their environment to obtain better conditions. To change unwanted unhealthy outcomes into desired healthy states may be seen as the ultimate goal of human manipulation. In this regard, the possibility to apply mathematical controllability theory to influence complex systems is extremely attractive.

This dissertation aims to bring insights by applying mathematical controllability theory to biological networks in an original way. In particular, we intend to study the importance of different driver nodes in controlling a context-specific target set, and to test the theory in empirical settings. We study two applications: the activation of macrophage cells, and the aging of the brain.

Recently, complex networks have received more and more attention and have been studied in disparate fields such as finance [START_REF] Bartolucci | A percolation model for the emergence of the Bitcoin Lightning Network[END_REF][START_REF] Delpini | Portfolio diversification, differentiation and the robustness of holdings networks[END_REF][START_REF] Ramadiah | Reconstructing and stress testing credit networks[END_REF], social sciences [START_REF] Hedayatifar | US social fragmentation at multiple scales[END_REF][START_REF] Kosack | Functional structures of US state governments[END_REF][START_REF] Power | Collective ritual and social support networks in rural South India[END_REF], ecology [START_REF] Domínguez-García | Unveiling dimensions of stability in complex ecological networks[END_REF][START_REF] Pilosof | The multilayer nature of ecological networks[END_REF][START_REF] Williams | Resistance and robustness of the global coral-symbiont network[END_REF], machine learning [START_REF] Ghasemian | Evaluating Overfit and Underfit in Models of Network Community Structure[END_REF][START_REF] Gray | Bayesian Inference of Network Structure From Information Cascades[END_REF][START_REF] Torres | GLEE: Geometric Laplacian Eigenmap Embedding[END_REF], epidemiology [START_REF] Colombi | Long-range movements coupled with heterogeneous incubation period sustain dog rabies at the national scale in Africa[END_REF][START_REF] Gilbert | Preparedness and vulnerability of African countries against importations of COVID-19: A modelling study[END_REF][START_REF] Oliver | Mobile phone data for informing public health actions across the COVID-19 pandemic life cycle[END_REF]. This attention led to multidisciplinary efforts that made possible a rich exchange and cross-contamination of techniques from different fields. One example is the application of controllability, from linear systems theory, to complex networks [START_REF] Kim | Role of graph architecture in controlling dynamical networks with applications to neural systems[END_REF][START_REF] Liu | Control Principles of Complex Networks[END_REF].

Roughly at the same time, in biology there has been an effort to go from reductionist approaches to integration and systems biology. This context promoted the representation of complex biological systems as networks [START_REF] Orphanides | A unified theory of gene expression[END_REF]. To apply controllability theory to biological networks was then a natural step forward [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Liu | Controllability of complex networks[END_REF].

In control theory, it is said that a system is controllable if it can be steered from any initial state to any final state in finite time, by applying an external input to well-chosen nodes, that are called drivers [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Luenberger | Introduction to Dynamic Systems: Theory, Models, and Applications[END_REF][START_REF] Slotine | Applied Nonlinear Control[END_REF]. When the external input is applied to only one driver node, the controllability problem is called single-input. If the interest is to change the state of a subset of nodes, instead of that of the whole system, we call this subset target set. Currently, biological network controllability focuses on two main directions. The first one is to find the minimum inputs required to control the whole or part of the system [START_REF] Liu | Controllability of complex networks[END_REF]. An issue with this approach is that in general many different sets of driver nodes can control the system [START_REF] Uno | Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs[END_REF]. The second direction is to evaluate the power of a single node in controlling the whole system [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF]. In this case, it is relatively easy to compute how many nodes can be controlled by a single driver, but it is not trivial to identify which subsets of nodes can be controlled simultaneously.

Moreover, the problem of evaluating a single driver in controlling a target set has surprisingly received little attention. It will be one of the main points addressed in this thesis.

From an empirical and biological perspective, it is often of interest to manipulate the state of only a subset of the nodes in the network [START_REF] Yan | Network control principles predict neuron function in the Caenorhabditis elegans connectome[END_REF], and it may be extremely difficult to act on more than one node at a time, as in the case of gene knockout, and brain stimulation. Our main research question is to understand what insight can controllability theory bring to biological networks, and in particular what is the role of a single node in influencing a context-specific target set of the network.

We tested our framework in two different biological applications: the activation of macrophages, and the aging of the brain.

We integrated network theory, linear systems theory, and biological knowledge to study the role of single driver nodes in controlling a context-specific target set, devising a measurable way to distinguish between a priori equivalent solutions. The main contribution of this dissertation is the development of a heuristic that we call step-wise target controllability [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF]. This heuristic is shown to be efficient for sparse directed networks. Our framework is original in the fact that it is not centered on the identification of drivers and targets, but instead on finding hidden functional relations between drivers and their controllable targets.

We used methods from linear system theory, thus we required a linear time invariant setting, in which the dynamics of the system is linear and the structure of the underlying network does not change in time.

We developed a heuristic stepwise algorithm that addresses the single input target control problem through an exact method that takes into account biological knowledge of the specific setting to lower the combinatorial complexity of the problem. We measured the importance of a driver node as the number of targets it can control. Additionally, we exploited context-specific information to assign an ordering to the target set, which made possible the identification of the highest-ranking controllable configuration of targets.

In the case of macrophage activation, we studied function through gene expression, and in the case of the brain, we adopted functional magnetic resonance imaging measurements.

We showed that this method is able to uncover hidden functional relations between drivers and their controllable targets. In the case of macrophage cells activation, we can identify dysregulated pathways and the genes that may be responsible for it. In the brain, we can characterize normal aging.

The dissertation proceeds as follows. Chapter 2

Network controllability: from engineering to biological systems

Network science is well-versed in multidisciplinary studies because of its ability to model the interaction of distinct units at different scales and levels of detail. Neural networks, transportation networks, electric power grids, the Internet, the World Wide Web, scientific coautorship and citations, genetic, metabolic, and protein networks are only some of the possible applications. Recently, there has been a growing effort in conducting multi-disciplinary projects, in the hope to achieve better understanding from different perspectives. This dissertation integrates multiple disciplines and in the following there will be a description of the fundamental concepts used.

In particular, it is described how mathematical controllability theory -motivated by biological questions -was applied to networks in general, and to biological networks.

Network science started as a branch of discrete mathematics, called graph theory.

We can say that it was born in 1736 when Euler solved the Königsberg's bridges problem. Since then there have been great advances in the study of network topology, and in the study of more and more complex systems. In general, we call a network complex when it presents some of the following qualities: it is large-scale, it has a higher-order dynamics, and it has an irregular connection pattern. In 1960, Erdös

and Rényi [START_REF] Erdös | On the evolution of random graphs[END_REF] proposed the concept of random graphs. More recently, Watts and Strogatz [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] studied the topology of not purely random neither purely regular networks, developing the concept of small world graphs. One year later, Barabási

and Albert [START_REF] Barabási | Emergence of Scaling in Random Networks[END_REF] proposed the scale-free model, able to explain the important role played by high-degree nodes, the hubs.

Chapter 2. Network controllability: from engineering to biological systems 5

The attention of the physics community and improved computational tools made possible the study of structural and topological network properties. Universal and unifying principles and statistical properties of graphs were identified [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF]. The debate then shifted to growth models, because the evolution of a network and its change in time (caused by external or internal forces) impose constraints and limitations to function and behavior [START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF]. Because of the strong link between topology and function, the attention shifted to the interactions between fundamental units of the network, asking if there could exist an interplay between network topology and dynamics, and if there could be common patterns in the dynamics [START_REF] Barzel | Universality in network dynamics[END_REF]. To answer these questions, perturbation studies and experiments were carried out. It was shown that perturbations of one node can alter the states of other nodes, due to the interaction between units [START_REF] Eguíluz | Scale-Free Brain Functional Networks[END_REF][START_REF] Kauffman | The ensemble approach to understand genetic regulatory networks[END_REF]. This property has been exploited to control a network.

Control theory is a branch of mathematics that studies the behavior of dynamical systems. The objective is to influence the state of the system by means of a feedback process: appropriately chosen inputs that are supposed to drive the the system to a target trajectory or final state. To bring the system's output to the desired one, a control action made of the difference between the actual and the desired output is applied as feedback to the input of the system [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. The property that determines if a system is controllable is called controllability. It informs on whether and on which conditions control may be achieved. We say that a system is controllable if it can be driven from any initial state to any final state in finite time, by applying inputs to well-chosen nodes in the system, called driver nodes [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Luenberger | Introduction to Dynamic Systems: Theory, Models, and Applications[END_REF]. Since controllability is a prerequisite for control, it is crucial to understand what factors determine it, in particular from a topological and structural perspective.

Mathematical control theory is widely applied in engineering systems. For instance in electric circuits, manufacturing processes, communication systems, aircraft and spacecraft, and coordination of moving robots. These physical systems can be mathematically described by means of a set of inputs, outputs, and state variables.

These elements are linked by a set of differential equations in the state space representation [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. Despite the fact that these equations are often nonlinear, control theory heavily relies on linear systems theory. This is mainly due to two factors.

First, controllability of locally linearized systems is studied before the full nonlinear problem, which is particularly accurate for systems near their equilibrium points.

Second, the complexity of the problem is increased by non-trivial network topologies, making it extremely difficult to handle a nonlinear setting. Instead, linear controllability can be seen as a prerequisite of nonlinear controllability. Moreover, Chapter 2. Network controllability: from engineering to biological systems 6

in some real problems such as consensus, the linear system represents an accurate descriptions and provides valid solutions [START_REF] Liu | Control Principles of Complex Networks[END_REF].

Linearization of nonlinear systems around and equilibrium point leads to a linear time invariant (LTI) system, where the structure of the system does not change in time. LTI systems are a sub-problem of linear systems. This dynamics was used to model for instance the amount of traffic that passes through a node on a communication network, or transcription factor concentration in a gene regulatory network.

A classical criterion to test for controllability in a LTI system for a given set of inputs is the Kalman rank condition, that relies on the controllability matrix of a system. It will be discussed in detail in Chapter 3.

Classical control problems typically take into account a single higher-dimensional system. In such problems the complexity may be given by high dimensionality, a stochastic setting, and a nonlinear dynamics. The difficulty of the problem makes

it not yet feasible to study a directed and irregularly connected network of many such systems. In the last decades there has been a shift in the focus of control problems, mainly motivated by technical reasons that a technological advance has brought forward. Rapid development of network science and engineering highly improved available resources such as supercomputers, huge databases, GPS services, and cloud computational environments, needed by new necessities from systems such as the Internet, the World Wide Web, wireless communication networks, power grids, global transportation systems, genome-scale metabolic networks, protein interaction networks, and gene regulatory networks [START_REF] Chen | Pinning control and synchronization on complex dynamical networks[END_REF]. Currently, the focus is on the study of many dynamical systems in a networked framework, where great attention is paid to directionality and connectivity. The complexity in this case may be given by the large number of interacting units, the high dimensionality, the sophisticated and irregular connection pattern, the nonlinear dynamics, the time-varying environment, the presence of multiple spatio-temporal scales.

In this setting, it is evident that it would be unfeasible and too expensive to control all the nodes in the network to achieve a certain desired objective. Wang and Chen [START_REF] Wang | Pinning control of scale-free dynamical networks[END_REF] and Li et al. [START_REF] Li | Pinning a complex dynamical network to its equilibrium[END_REF] introduced pinning control, a new strategy that takes into account both dynamical rules and network topology in control systems design and implementation. The aim of this strategy is to control the whole dynamical network by directly imposing controllers to only a fraction of selected nodes in the network. Pinning control was inspired and motivated by biology, in fact in nature it was observed that a small fraction of individuals is able to influence the behavior of a whole group [START_REF] Chen | Pinning control and synchronization on complex dynamical networks[END_REF]. This is true for instance for foraging bees and fish schools [START_REF] Couzin | Effective leadership and decision-making in animal groups on the move[END_REF].

This framework led to a new set of questions. In particular, it was asked how Chapter 2. Network controllability: from engineering to biological systems 7

many and which nodes in a network are needed to achieve the control goal most effectively. It is now clear that the answer to the above question depends on the structure of the network as well as on the dynamical laws governing it. Modern control theory studies the relation between control inputs and system state with a particular attention to the interplay between network topology and dynamics.

In general, we may say that to understand the behavior of a system we are required to measure its state variables and to model its dynamics [START_REF] Liu | Control Principles of Complex Networks[END_REF][START_REF] Liu | Controllability of complex networks[END_REF]. Ideally, we should perfectly know: i) the architecture of the system, that is its structure and interconnection pattern; ii) the dynamical laws governing it; and iii) all the system's parameters. In practice of course this is not the case, and we lack an accurate description of one or more of these three factors. For instance, it was estimated that in protein-protein interaction networks we are currently able to infer about 20% of all the plausible interactions [START_REF] Sahni | Widespread macromolecular interaction perturbations in human genetic disorders[END_REF]; an analytical form of the dynamics in communication systems is often missing; and accurate values of biochemical reaction systems are extremely difficult to retrieve [START_REF] Liu | Control Principles of Complex Networks[END_REF]. It is also true that recently network controllability greatly advanced despite the lack of this ideal and complete knowledge. This may have also been favored by the rich cross-contamination from other fields. A brilliant example of this is the case of structural controllability.

Structural controllability of complex networks was first introduced by Lin [START_REF] Lin | Structural controllability[END_REF];

it refers to the property of a system to be controllable for almost all of its numerical realizations [START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF], bringing attention to the topology. In particular, it evaluates the controllability of binary networks, in which edges are either present, or absent, effectively overcoming the need for estimating the system parameters. At the time when this concept was introduced, sufficient and necessary conditions to check whether a system with LTI dynamics was controllable were given. However, an algorithm to compute it was not provided.

Later, generic properties of structured systems were studied, allowing for a mapping of system's properties into graph theoretical conditions [START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF][START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF][START_REF] Van Waarde | A Distance-Based Approach to Strong Target Control of Dynamical Networks[END_REF]. Liu et al.

[99] integrated knowledge from statistical physics, spin glass theory and network science to map the structural controllability problem into a well-known graph theoretical problem: the maximum cardinality matching. They were able to provide answers to questions such as: what is the minimum set of nodes that should be controlled to achieve control over the system? How to estimate the fraction of such driver nodes?

Notice that, despite the claim that topology and degree distribution play a major role in network controllability [START_REF] Liu | Controllability of complex networks[END_REF], other authors stressed the importance of also taking into account the nodal dynamics [START_REF] Cowan | Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks[END_REF][START_REF] Wang | A geometrical approach to control and controllability of nonlinear dynamical networks[END_REF][START_REF] Zhao | Intrinsic dynamics induce global symmetry in network controllability[END_REF]. Exact controllability crite-Chapter 2. Network controllability: from engineering to biological systems 8

ria such as the Kalman rank condition, the Popov-Belevitch-Hautus test, and the Gramian criterion -all discussed in Chapter 3 -are limited by shortcomings such as parameter estimation and ill-conditioned numerical problems. However, they continue to be investigated because -unlike the structural controllability frameworkthey allow for the computation of the required control energy and other metrics [START_REF] Pasqualetti | Controllability Metrics, Limitations and Algorithms for Complex Networks[END_REF].

In some settings, it may not be feasible to control the whole network. For instance, it was predicted that for gene regulatory networks over 80% of the nodes should be drivers in order to achieve control over the system [START_REF] Kanhaiya | Controlling Directed Protein Interaction Networks in Cancer[END_REF][START_REF] Liu | Controllability of complex networks[END_REF]. Moreover, in some applications it is not even necessary to control the whole network. This may be the case in smart grids, for the installation of controllers and for choosing the locations of actuators with physical limitations; and in the development of drug cocktails for multiple drug target identification in gene regulatory networks [START_REF] Wu | Network Output Controllability-Based Method for Drug Target Identification[END_REF][START_REF] Yang | Finding multiple target optimal intervention in disease-related molecular network[END_REF].

To describe such problems, we can define a target control problem in which the aim is to find the minimum number of driver nodes required to steer the state of only a selected subset of the network's nodes, a subsystem. These nodes are called target nodes, and are chosen because they are supposedly essential for a certain task or mission of the system. The target control problem for linear networks is a particular case of output control, and a generalization of the full control problem, which requires the control over the entire system [START_REF] Czeizler | Structural Target Controllability of Linear Networks[END_REF].

The multi-input target control problem was shown to be NP-hard [START_REF] Czeizler | Structural Target Controllability of Linear Networks[END_REF], thus heuristics and approximate solutions have been proposed to solve it. Among the firsts to recently address this problem, Gao et al. [START_REF] Gao | Target control of complex networks[END_REF] proposed a structural approach in the form of a greedy algorithm. Later, Czeizler et al. [START_REF] Czeizler | Structural Target Controllability of Linear Networks[END_REF] built on their work improving the algorithm. A different perspective on the problem was proposed by Klickstein et al. [START_REF] Klickstein | Energy scaling of targeted optimal control of complex networks[END_REF], who focused on the control energy, and implemented a target problem with the aim to reduce the energy, without changing the predesignated driver nodes; they studied the relation between control energy and cardinality of the target set. Variations of the problem such as strong structural controllability [START_REF] Van Waarde | A Distance-Based Approach to Strong Target Control of Dynamical Networks[END_REF], symmetric networks [START_REF] Li | Structural Target Controllability of Undirected Networks[END_REF], and functional target controllability [START_REF] Commault | Functional target controllability of networks: Structural properties and efficient algorithms[END_REF], where the interest is in controlling the whole trajectory instead of only the final state, were recently proposed. In all those cases, necessary and sufficient graph conditions for target controllability were provided.

It is important to notice that a trivial yet legit objection arises: would it be easier to directly control the target nodes? In practice, that may seldom be the case, in fact we may have a large and prohibitive number of targets, and the target nodes may not be directly accessible via a control action.

Chapter 2. Network controllability: from engineering to biological systems 9

The main goal of the control problems discussed so far was to identify a minimum driver node set (MDS) able to achieve control over all or part of a system. It had been shown that the minimum number of drivers required for control, that is the cardinality of the MDS is fixed [START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF][START_REF] Liu | Controllability of complex networks[END_REF], however the individual nodes in the set may vary [START_REF] Jia | Control Capacity and A Random Sampling Method in Exploring Controllability of Complex Networks[END_REF][START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF][START_REF] Klickstein | Energy scaling of targeted optimal control of complex networks[END_REF].

It remains an open question to enumerate all possible minimum driver node sets, largely due to the huge number of maximum cardinality matchings [START_REF] Uno | Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs[END_REF], even in simple graphs such as trees [START_REF] Heuberger | The number of maximum matchings in a tree[END_REF][START_REF] Wagner | On the number of matchings of a tree[END_REF], bipartite graphs [START_REF] Liu | Number of maximum matchings of bipartite graphs with positive surplus[END_REF], or random graphs [START_REF] Zdeborová | The number of matchings in random graphs[END_REF]. It is important to notice that since the MDS is not unique, nodes may not participate in control equally, prompting works that quantify their participation [START_REF] Commault | Sensor classification for the fault detection and isolation, a structural approach[END_REF][START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF][START_REF] Jia | Control Capacity and A Random Sampling Method in Exploring Controllability of Complex Networks[END_REF][START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF]. This observation motivated the investigation of the composition of different MDSs, and the classification of the role of individual driver nodes in exercising control.

Commault et al. [START_REF] Commault | Observability Preservation Under Sensor Failure[END_REF] worked on a classification for observability -the dual of controllability -and later extended their results to account for controllability with the hypothesis that each driver receives an independent input, also allowing for a case in which not all nodes can directly receive a control input [START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF]. Driver nodes were classified as essential, if they appeared in all configurations; useful, if they appeared in some configurations; and useless, if they were never present in a configuration.

Jia and Barabási [START_REF] Jia | Control Capacity and A Random Sampling Method in Exploring Controllability of Complex Networks[END_REF] studied the case in which a single input signal can be applied to multiple nodes. They devised a classification based on the likelihood of each driver node to participate in a MDS. They estimated this probability with a random sampling algorithm, nodes that are always (never) present in the MDSs are called critical (redundant), and ordinary if they belong to the set sometimes.

Participation in a MDS could give insight into the role of a driver when sharing control with other nodes, but it does not answer to a more fundamental question of its role in controlling the system in the case in which it is the only driver node.

Typically in network theory the importance of a node is computed by means of centrality measures based on topology. Some examples are the degree, the betweennes, and the eigenvector centrality [START_REF] Newman | Networks: An Introduction[END_REF]. A measure of centrality based on controllability theory was proposed by Liu et al. [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF]. Their control centrality is quantified as the size of the controllable subspace when a certain driver node is considered, and it can be computed through an optimization problem.

The single-input case may be particularly useful in biological contexts, where it could be unfeasible or extremely difficult to act on multiple driver nodes.

Gu et al. [START_REF] Gu | Controllability of structural brain networks[END_REF] studied structural brain networks, and for single brain areas proposed multiple measures of importance based on the Gramian matrix (defined in Chapter 3).

Chapter 2. Network controllability: from engineering to biological systems 10

Recent developments in network theory, together with advances in computational tools and medical techniques made possible the study of biological systems from new, more detailed perspectives. Moreover, there has been a growing interest in applying control theory to biological systems. This was partially motivated by the intuitive fact that control and regulation are key-factors for biological systems [START_REF] Vecchio | Biomolecular Feedback Systems[END_REF][START_REF] Iglesias | Control Theory and Systems Biology[END_REF], which -for instance -are able to maintain homeostasis, sensing and processing external and internal cues in changing environmental conditions. In the evolution of molecular interaction networks, it may be particularly relevant the need to implement control in the form of input signals steering the network to specific functional states [START_REF] Vinayagam | Controllability analysis of the directed human protein interaction network identifies disease genes and drug targets[END_REF].

At a cellular level, control theory was applied to biological systems in the form of experimental and computational approaches that made possible the regulation of protein expression in a population of yeast cells [START_REF] Menolascina | In-Vivo Real-Time Control of Protein Expression from Endogenous and Synthetic Gene Networks[END_REF], and to maintain a bistable genetic circuit to unstable equilibrium [START_REF] Lugagne | Balancing a genetic toggle switch by real-time feedback control and periodic forcing[END_REF].

At large scale, such detailed control is not yet possible. However, recent studies were able to apply controllability theory to different kinds of biological networks, gaining substantial insight. In protein-protein interaction networks, Vinayagam et al. [START_REF] Vinayagam | Controllability analysis of the directed human protein interaction network identifies disease genes and drug targets[END_REF] exploited controllability theory to identify drug targets; Ravindran et al. [START_REF] Ravindran | Network controllability analysis of intracellular signalling reveals viruses are actively controlling molecular systems[END_REF] studied the virus-host interaction networks to understand the virus exploitation of the host system; Ravindran et al. [START_REF] Ravindran | Identification of critical regulatory genes in cancer signaling network using controllability analysis[END_REF] analyzed the cancer signaling network to propose therapeutic targets. Moreover, Yan et al. [START_REF] Yan | Network control principles predict neuron function in the Caenorhabditis elegans connectome[END_REF] performed experimental studies to test the predictions of controllability theory in the case of individual neurons and neuronal classes in the locomotor behavior of the C. Elegans nematode.

Studying the controllability of biological systems, a particular attention has been paid to the brain, a complex system that operates at different spatial and temporal scales. It enables humans to perform complex cognitive functions trough the alteration of the dynamics of neural systems in order to meet task demands. These capabilities are referred to as cognitive control [START_REF] Botvinick | Motivation and Cognitive Control: From Behavior to Neural Mechanism[END_REF][START_REF] Heatherton | Cognitive neuroscience of self-regulation failure[END_REF], and cognitive function is driven by dynamical interactions between large-scale neural circuits, enabling behavior.

At a system-level, advances in network theory and in the study of the topological properties of brain networks involve questions concerning the relation between structure and function, and how the evolution of the architecture of the brain is reflected in its functioning [START_REF] Fornito | Fundamentals of Brain Network Analysis[END_REF]. From a control-theoretical perspective, researchers focused on studying the controllability of the brain, and the brain areas that are most influential in driving changes to brain dynamics. Studies of non-invasive transcranial magnetic stimulation [START_REF] Bikson | Safety of Transcranial Direct Current Stimulation: Evidence Based Update 2016[END_REF] and invasive deep brain stimulation [START_REF] Wilson | Clustered Desynchronization from High-Frequency Deep Brain Stimulation[END_REF] were Chapter 2. Network controllability: from engineering to biological systems 11

applied to healthy and diseased subjects, and are used for clinical treatments, as in the case of Parkinson's and Alzheimer diseases, epilepsy and stroke [START_REF] Muldoon | Stimulation-Based Control of Dynamic Brain Networks[END_REF].

In conclusion, advances in network theory computational tools, and available resources, motivated by biological questions, prompted the study of biological systems from a networked, control-theoretical perspective.

Chapter 3

State-of-the-art on network controllability

We wish to understand what insight can mathematical controllability theory bring to biological networks, to study the importance of single driver nodes in controlling a context-specific target set, and to test the theory in empirical settings. To do so, we devised a measure of importance of the driver nodes, analyzing in an original way the control they may exert on target nodes in two biological applications: the activation of macrophage cells, and the aging brain.

In this chapter, we report fundamental notions of graph theory and controllability theory.

Graph theory

Graph theory is the natural framework for the mathematical treatment of complex networks [START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF]. Many objects of interest in the physical, biological, and social sciences can be modeled as networks and thinking of them in this way can lead to new and useful insights [START_REF] Fornito | Fundamentals of Brain Network Analysis[END_REF][START_REF] Newman | Networks: An Introduction[END_REF].

A graph G = (V, E ) consists of two sets V and E, such that V = ∅ and E is a set of either unordered for undirected graphs, or ordered for directed graphs, pairs of elements of V. The elements of V ≡ {v 1 , ..., v N } are the nodes, or vertices, of the graph G, while the elements of E ≡ {l 1 , ..., l E } are its links, or edges.

A weighted graph is a graph in which a number, the weight, is assigned to each edge. A binary, or unweighted, graph only accounts for the existence of the connection, and not for its strength.

In a undirected, or symmetric, graph, each of the edges is defined by a pair of nodes i and j , and is denoted as (i, j) or l ij . The edge is said to be incident in nodes i and j, or to join the two nodes; the two nodes i and j are referred to as the end-nodes of link (i, j).

In a directed graph, the order of the two nodes is important: l ij stands for a link from j to i , and l ij = l ji . Edge l ij is said to be incident in node i; node i is the end-nodes of link (i, j).

Two nodes joined by a link are referred to as adjacent or neighboring.

An isolated or disconnected node is a node that has no neighbors.

The adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of nodes are connected or not in the graph. The elements of the adjacency matrix are either zeros and ones, for binary networks, or real numbers, for weighted networks. If the graph is undirected, the adjacency matrix is symmetric.

We call a walk a set of edges in an undirected graph that is ordered to form a sequence in which any pair of successive edges shares a common node. In a directed graph, the definition is analogous, but with the added restriction that the edges have to be directed in the same direction. A walk can traverse multiple edges and nodes, visiting the same node or edge on multiple occasions.

A closed walk, or cycle, is a walk in which the first and last nodes are the same.

A graph is said to be acyclic if it does not contain any cycle.

A path is a walk in which all edges and nodes are unique.

We call a tree an undirected graph in which any two nodes are connected by exactly one path [START_REF] Newman | Networks: An Introduction[END_REF]. A rooted directed tree, or tree, is a directed graph in which, for a vertex i called the root and any other vertex j, there is exactly one directed path from i to j.

A subgraph of a graph G is another graph formed from a subset of the vertices and edges of G. The vertex subset must include all endpoints of the edge subset, but may also include additional vertices.

A graph is said to be connected when there is a path between every pair of nodes, that is there is a sequence of edges which connects a sequence of every two vertices in the graph.

A Global scale properties usually study the efficiency of information transfer within the network. For instance, global-efficiency represents a measure of integration, and local-efficiency a measure of segregation [START_REF] Fornito | Fundamentals of Brain Network Analysis[END_REF][START_REF] Gonzalez-Astudillo | Network-based brain computer interfaces: Principles and applications[END_REF].

Controllability theory

Mathematical control theory is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems.

To control an object means to influence its behavior so as to achieve a desired goal, the central tool is the use of feedback in order to correct for deviations from the desired behavior [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. Controllability is a prerequisite for control.

The controllability property of a system refers to the fact that any initial state can be steered to any final state by choosing the inputs appropriately, the intuition behind this idea is represented in Figure 3.1.

Controllability of a system [92]

A state determined dynamic system defines, for every pair of initial and final times, n 0 and n 1 , a mapping of the initial state

x(n 0 ) and an input trajectory u(n), n ∈ [n 0 , n 1 ], to the state at time n 1 , x(n 1 ). We shall use the notation s (n 1 , n 0 , x(n 0 ), u(•)) to denote this state transition mapping, i.e. x(n 1 ) = s (n 1 , n 0 , x(n 0 ), u(•)) if the initial state at time n 0 is x(n 0 ), and the input trajectory is given by u(•).

Consider a state determined dynamical system with a transition map s (n 1 , n 0 , x 0 , u(•)) such that

x(n 1 ) = s (n 1 , n 0 , x(n 0 ), u(•)) (3.1)
The dynamical system is said controllable on

[n 0 , n 1 ] if ∀x 0 , x 1 , ∃u(•) ∈ U such that s (n 1 , n 0 , x 0 , u(•)) = x 1 (3.
2)

The dynamical system is said controllable at n 0 if ∀x 0 , x 1 , ∃n 1 ≥ n 0 and u(•) ∈ U such that s (n 1 , n 0 , x 0 , u(•)) = x 1 (3.3) Continuous time linear system Consider the continuous time linear system

with n ∈ R ẋ(n) = A(n)x(n) + B(n)u(n) (3.4)
We study the case in which matrices A(n) and B(n) are independent on time (A, B), called linear time invariant (LTI). Equation 3.4 becomes

ẋ(n) = Ax(n) + Bu(n) (3.5)
The vector x(n) ∈ R N captures the state of a network of N nodes at time n. For example, x i (n) can denote the amount of traffic that passes through a node i in a communication network [START_REF] Pastor-Satorras | Evolution and Structure of the Internet: A Statistical Physics Approach[END_REF], transcription factor concentration in a gene regulatory network [START_REF] Lezon | Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns[END_REF], the magnitude of neurophysiological activity in brain networks [START_REF] Gu | Controllability of structural brain networks[END_REF], or the gene expression in a gene coexpression network.

The N × N matrix A is the state matrix which describes the structure of the system. It is the adjacency matrix of the network and describes the system wiring diagram and the interaction strength between the components, for example the traffic on individual communication links, the strength of a regulatory interaction, the number of white matter streamlines connecting brain regions, or the significant correlations among gene expressions.

Finally, B is the N × M input matrix (M ≤ N ) that identifies the nodes controlled by an outside controller. The system is controlled using the time-dependent input vector u(n) ∈ R M imposed by the controller. Despite the fact that in general the same signal u i (n) can drive multiple nodes, we will consider only the case in which each signal controls exactly one node. This hypothesis leads to a matrix B such that each row and column contains only one nonzero element, corresponding to the node that will receive the external input u(n).

Nodes identified by matrix B when the system is controllable are called driver nodes. Notice that there may be multiple sets of driver nodes able to achieve control over the system.

If we apply the external input to a single driver node, M = 1, the matrix B becomes a column vector, and we call the problem single-input.

The single-input setting allows us to exploit the definition of control centrality as in Liu et al. [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF]. Mathematically, control centrality of node i captures the dimension of the controllable subspace or the size of the controllable subsystem when we control node i only.

Note that to identify and analyze the control input u(n) is outside the scope of this work.

Controllability criteria for LTI systems

Classically, there are three main equivalent criteria to test for controllability in a LTI system. Those are the Kalman criterion [START_REF] Dorf | Modern Control Systems[END_REF][START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF], the Popov-Belevitch-Hautus criterion [START_REF] Camlibel | Popov-Belevitch-Hautus type controllability tests for linear complementarity systems[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF], and the Gramian criterion [START_REF] Chen | Linear System Theory and Design, International 3rd. Edition[END_REF].

Kalman criterion

The Kalman criterion, also known as the Kalman rank condition, is a necessary and sufficient condition for the controllability of a system [START_REF] Dorf | Modern Control Systems[END_REF][START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF]. Consider a LTI system as in Equation 3.5, then define the N × N M controllability matrix Q as

Q = B AB A 2 B • • • A N -1 B (3.6)
We say that a system is controllable if the controllability matrix has full row rank

rank(Q) = N (3.7)
That is, if the system is controllable, Q will have N columns that are linearly independent, then every state is reachable by giving the system proper inputs. This criterion can be implemented for any kind of graph: binary or weighted, directed or symmetric.

Trivially, when all nodes receive an external input, B becomes the identity matrix of order N , thus Q is always full rank and the system is controllable.

In the Kalman rank condition, for any

N > N , rank B AB • • • A N -1 B = rank B AB • • • A N -1 B = rank(Q) (3.8)
Thus, only powers of A up to N are required to test for controllability.

We notice that if the weights in the adjacency matrix are all equal, for instance all ones, this may be seen as a worst-case scenario for controllability. In fact, it is more likely for two rows to be linearly dependent compared to the case with real weights.

There are some limitations to the implementation of the Kalman rank condition.

First of all -as the two other equivalent criteria -it is a brute-force approach, that has exponential complexity if requested to test all possible numbers of driver nodes and binomial complexity for a fixed number of driver nodes.

In fact, for a fixed number of drivers M , 1 ≤ M ≤ N , there are ( N M ) different configurations of the matrix B, in fact we choose M ones representing the drivers among the N columns, and the complexity can be computed as N m=1 ( N m ) = 2 N -1. Along with the exponential complexity in time, the problem of executing an accurate rank test is ill-conditioned and it may be sensitive to roundoff errors and uncertainties in the matrix elements [START_REF] Liu | Control Principles of Complex Networks[END_REF].

Interpretation of the controllability matrix in the single-input case

The controllability matrix Q is made of a linear combination of the column vector B and powers of A. For a binary adjacency matrix, the element [A k ] ij can be interpreted as the number of walks of length k between each pair of nodes, from node j to node i (see [START_REF] Newman | Networks: An Introduction[END_REF], p.136).

Then, in the single-input case, the interpretation of the elements Q ij is as fol-lows. The first column Q i1 corresponds to the nodes receiving the input B, then elements on the second column Q i2 represents the number of paths of length 1 from the driver to the other nodes i. In general, Q ij is the number of paths of length j -1 from the driver to the other nodes in the network.

Popov-Belevitch-Hautus (PBH) criterion

This criterion presents a condition in terms of the eigenmodes of the system [START_REF] Camlibel | Popov-Belevitch-Hautus type controllability tests for linear complementarity systems[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. Consider a LTI system, as in Equation 3.5, then the system is controllable if and only if

λ ∈ C, z ∈ C N , z * A = λz * , z * B ≥ 0 ⇒ z = 0 (3.9)
It is also possible to express the PBH test saying that the system is controllable if and only if it holds rank (λI

N -A, B) = N ∀λ ∈ C (3.10)
where I N is the identity matrix of order N .

Note that rank (λI N -A, B) is less than N only if λ is an eigenvalue of A. This is also equivalent to say that there is no left eigenvector of A that is a left zero vector of B.

Gramian criterion

The Gramian controllability criterion states that a system is controllable if and only if its controllability Gramian is full rank [START_REF] Chen | Linear System Theory and Design, International 3rd. Edition[END_REF][START_REF] Li | Lecture notes for the course 'Advanced Control System Design[END_REF].

An alternative representation of the continuous system in Equation 3.4 is given by

x(n 1 ) = φ(n 1 , n 0 )x n 0 + n 1 n 0 φ(n 1 , n)B(n)u(n)dn (3.11)
Then, the controllability Gramian on the time interval [n 0 , n 1 ] for a LTI continuous system is defined as

W [n 0 ,n 1 ] = n 1 n 0 e A(n 0 -n) BB T e A T (n 0 -n) dn (3.12)
where B T and A T stand for the transposed matrices of B and A respectively.

The system is controllable if its controllability Gramian is full rank on the time

interval [n 0 , n 1 ].
In discrete time, the controllability Gramian on the time interval [n 0 , n 1 ] of a LTI system is defined as

W [n 0 ,n 1 ] = n 1 -1 j=n 0 A j BB T (A T ) j (3.13)
Analogously, the system is controllable if its controllability Gramian is full rank on the time interval [n 0 , n 1 ].

Structural controllability

Structural controllability was introduced to give more emphasis to the topology of networked systems and to overcome known issues of classical controllability criteria, such as the difficulty of parameter estimation and the occurrence of numerical errors in the rank tests computations.

The objects of study of structural controllability are structured matrices, in which it is assumed that the elements are either zeros, or independent parameters. These matrices describe binary networks.

Let A and B be structured matrices with a LTI dynamics (Equation 3.5).

Then we say that the system described by (A, B) is structurally controllable if it is possible to choose the non-zero weights in A and B such that the system satisfies the Kalman rank condition. A structurally controllable system can be shown to be controllable for almost all weight combinations, except for some pathological cases with zero measure [START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF][START_REF] Lin | Structural controllability[END_REF].

The main advantage of structural controllability is that necessary and sufficient conditions for controllability can be provided as graph-theoretical conditions [START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF][START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF][START_REF] Van Waarde | A Distance-Based Approach to Strong Target Control of Dynamical Networks[END_REF].

Maximum cardinality matching

One recent important result from Liu et al.

[99] was to map the Kalman rank condition (Equation 3.7) into the maximum cardinality matching over a graph.

A matching in a graph G = (V, E ) is a subset M of the edges E such that no two edges in M share a common end node.

A maximum cardinality matching is a matching that contains the largest possible number of edges.

A perfect matching occurs when all nodes are matched in a symmetric graph, or all but one nodes are matched in a directed graph.

A node is matched if it is an endpoint of one of the edges in the matching M, otherwise the node is unmatched. In particular, a matched edge in an undirected graph, will produce two matched nodes, both nodes to which it is incident. In a directed graph a matched edge will produce only one matched node.

In the framework of Liu et al. [START_REF] Liu | Controllability of complex networks[END_REF], given a maximum cardinality matching, the minimum number of inputs required to achieve control over the system is one if there is a perfect matching, and is equivalent to the number of unmatched nodes otherwise. Then, the driver nodes correspond to the unmatched nodes, or to any single node in the case of perfect matching.

Instead of a brute-force test for controllability, this framework provides us with a method to identify a configuration of driver nodes. The maximum cardinality matching is a NP-hard problem if required to find all the feasible configurations, but since directed graphs can be mapped in symmetric bipartite graphs, heuristics exist to find one configuration in polynomial time [START_REF] Hopcroft | A n5/2 algorithm for maximum matchings in bipartite[END_REF].

Limitations One limitation of this approach is that, by definition, some categories of matrices cannot be considered structured because their non-zero parameters are not independent. Recent studies are working to overcome this issue in the case of symmetric matrices [START_REF] Li | Structural Target Controllability of Undirected Networks[END_REF][START_REF] Menara | Structural Controllability of Symmetric Networks[END_REF].

In the structural controllability framework it is not possible to give an estimate of the control energy required to achieve a certain task. In fact the answer to the controllability test is binary: either a system is controllable or not, and there are no cues to the difficulty of the problem.

In general, in the structural controllability framework (as in [START_REF] Liu | Controllability of complex networks[END_REF]), it is assumed that the same input can be applied to multiple driver nodes. However this is not always a realistic assumption, and it may not be feasible in some practical cases [START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF].

With these limitations in mind, we developed an exact method to study the role of single driver nodes in controlling a targeted system.

Chapter 4

Step-wise target controllability

Target controllability

The controllability property of a system refers to the ability to affect the state of the whole network. In practice however, we may be interested in modifying the states of a selected subset of nodes, the target nodes. If our goal is to control a subsystem, it is possible to accommodate this constraint in the Kalman criterion, following the framework of [START_REF] Gao | Target control of complex networks[END_REF], and the recent application in [START_REF] Yan | Network control principles predict neuron function in the Caenorhabditis elegans connectome[END_REF].

Let T be the target set, the subset of nodes to be controlled, of cardinality |T | = S. We introduce the S × N matrix C whose rows are rows of the identity matrix of order S, with the non-zero element in correspondence with a target node.

Then, we can define the analogous of Equation 3.6, the target controllability matrix

Q T = C × Q = CB CAB CA 2 B • • • CA N -1 B (4.1)
And we can update the Kalman rank condition, Equation 3.7. The driver nodes identified in B can control the target set T if and only if

rank(Q T ) = S (4.2)
Note that when T contains only one target node, S = 1, then the target is controllable only if it is reachable from the driver nodes. In fact the matrix C will have one row, and since rank(C × Q) takes into account only the row of Q corresponding to the target node, there would be at least a non-zero element only if there is at least one path from the drivers to the target.

Interpretation of the controllability matrix in the single-input target case

In the controllability matrix computation of the single-input problem, B filters out columns of A k in such a way that only those corresponding to the driver node are retained, Section 3.2.1. In addition, in the single-input target problem, the matrix C filters out rows of Q in such a way that only those corresponding to target nodes are retained. As a result, for a binary adjacency matrix, elements of Q T = C × Q represent the number of walks starting from the driver (due to matrix B) and arriving in the targets (due to matrix C).

Step-wise target controllability

Linear systems controllability theory has a wide range of applications and a high versatility. It mainly deals with the problem of identifying the nodes in which to apply suitable inputs such that a system can be steered from an initial state to a desired final state. It can be applied to biological systems, where it is often the case that additional constraints need to be enforced. In particular, a context-specific subsystem may be of interest, and it may not be feasible to simultaneously act on more than one input. These two additional constraints lead to what's called a single-input target control problem.

We addressed this specific problem, with the aim to gain insight from the application of controllability theory to biological systems, and in particular to study what is the role of a single driver node in controlling a target set.

Notice that computing the rank of the controllability matrix is ill-conditioned for a large network, thus it was not feasible for us to test all target nodes at the same time. To overcome this issue, we devised ways to lower the combinatorial complexity of the problem, working on smaller subsystems.

Given a target set T , and a single driver node, we imposed an ordering to the elements of T , and we developed a strategy to identify the highest-ranked controllable configuration of targets. We call this procedure step-wise target controllability.

This strategy implements the Kalman rank condition in an original way, removing the nodes known to be irrelevant to the constrained problem.

The ordering of the target set is required to lower the combinatorial complexity of the problem. In fact, in general the Kalman criterion returns the number of controllable targets, and there may be up to a factorial number of a priori equivalent controllable configurations of targets.

If |T | = S target nodes are of interest, but only rank(Q T ) = S 1 of them are controllable, a priori there is a factorial number, ( S S 1 ), of possible controllable con- figurations.

Instead, the ordering of the target set allows us to pinpoint the unique configu-ration with the highest-ranked targets.

It is trivial to notice that there is a single controllable configuration when all target nodes can be controlled.

Remove non necessary nodes from the computation

Consider a single-input target control problem, in which there is one driver node (B is a column vector) and a set of target nodes T . Because of the interpretation of the target controllability matrix Q T , not all the nodes in the network are required for the computation of the rank condition.

In particular, i) because of the presence of B, only nodes accessible from the driver are required, and ii) among those, only nodes that are on a path from the driver to the targets need to be considered.

Rows and column not corresponding to the above cases will provide all-zero contribution to the target controllability matrix Q T , because there are no paths that from the driver reach those intermediate nodes and then the targets. Thus, they would not contribute to the rank and can safely be discarded from the computation.

Removing those nodes lowers the occurrence of numerical errors because manipulating a sub-system of N 1 < N nodes only requires to compute powers of A up to N 1 -1 to form the controllability matrix, Equation 3.6.

The algorithm

The step-wise target controllability strategy mainly takes advantage of two keyconcepts: the ordering of the target set, and the fact that a controllable configuration of target nodes can trivially be identified when all targets are controllable. The strategy can be described by a simple algorithm.

Given an ordered target set T , each node in the network is tested as a driver.

We compute the step-wise target control centrality (τ ) of a driver as the number of targets that it can control. Let [t 1 , ..., t S ] be the ordered sequence of target nodes, T is the current target set and contains the target nodes we are currently testing, τ * is the number of controllable targets among the ones already tested. We compute the step-wise target control centrality as follows.

• Step 0 : the target set contains the first target node, T = {t 1 }, the initial step-wise target control centrality is zero, τ * = 0.

• Step 1 : build the subgraph G , made of nodes accessible from the driver that can reach the nodes in the target set T .

• Step 2 : perform the Kalman criterion to check for target controllability of the subgraph G .

-If the configuration is not controllable, discard the last target added to the target set and test the successive one. At a general step when the last node added to T was t j , it means T = T \ {t j } and T = T ∪ {t j+1 }.

-Else, include the successive target in the target set and increase by one the step-wise target control centrality. At a general step when the last node added to T was t j , it means T = T ∪ {t j+1 } and τ * = τ * + 1.

• Repeat steps 1 and 2 until the last target node is tested.

At the end of this procedure, the step-wise target control centrality of the driver currently being tested is τ = τ * , and T contains its highest-ranked controllable configuration of targets.

Toy example

Functioning of the step-wise target controllability framework can be appreciated in the following example. Let us consider a directed full binary tree with h = 6 levels, with the root node as the candidate driver. Without loss of generality, we randomly position a target in each level and we rank them according to their height in the tree. Then, we introduce a simple cycle among the first three nodes of the tree, as in Figure 4.1. By construction, this configuration is controllable and the entire target set can be fully driven by the driver [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF]. However, when computing the rank with Matlab (https://www.mathworks.com/) rank() function considering the entire network, the returned rank is deficient. Instead, by removing the part of the network that is irrelevant for controllability, the rank is full and we retrieve the entire controllable configuration, even in the case of larger networks, i.e. up to h = 10 levels.

Limitations and possible developments

The step-wise target controllability framework can be implemented for any network, since the Kalman rank condition on which is based is a general criterion. However, our framework is mainly beneficial to sparse, directed networks. In fact, for this kind of networks the dimensionality reduction has a real impact on the computation.

Instead, in the case of undirected graphs it is not possible to find nodes that are not on a walk from the driver to the targets since all nodes belong to the same connected component. Similarly, for directed networks, as the density increases We start from a simple directed full binary tree, and we add a cycle among the first three nodes. The driver is the root of the tree, while we put a target node in each level of the tree. Target nodes are then ranked according to their height h. This configuration is fully target controllable by construction regardless of the tree's height h. However, for h = 5 (i.e. N = 63), the standard procedure computing the rank of the full Kalman controllability matrix cannot retrieve all the targets, as the rank computation is deficient due to numerical errors. Instead, by using the step-wise target controllability we can correctly identify the controllable targets up to h = 10, i.e. N = 2047.

more and more nodes become part of the same strongly connected component, until the whole network is strongly connected.

It may be interesting to implement a similar step-wise procedure varying the central controllability test. For instance, in some cases the Popov-Belevitch-Hautus criterion may be more numerically stable, or, in the case of a nonlinear dynamics, a feedback vertex set [START_REF] Zañudo | Structure-based control of complex networks with nonlinear dynamics[END_REF] procedure may be implemented.

As for the choice of the target set, also the choice of an ordering is contextspecific, and should be considered for each particular application. For instance, targets can be sorted according to their importance in a biological function (e.g.

genes) [START_REF] Liseron-Monfils | NECorr, a Tool to Rank Gene Importance in Biological Processes using Molecular Networks and Transcriptome Data[END_REF][START_REF] Zhao | Ranking Candidate Disease Genes from Gene Expression and Protein Interaction: A Katz-Centrality Based Approach[END_REF] or in discriminating pathological conditions (e.g. brain areas) [START_REF] Lohmann | Eigenvector Centrality Mapping for Analyzing Connectivity Patterns in fMRI Data of the Human Brain[END_REF][START_REF] Sen | Ranking Regions, Edges and Classifying Tasks in Functional Brain Graphs by Sub-Graph Entropy[END_REF].

It should be noticed that in the step-wise target controllability framework, the procedure does not account for the case in which the ordering of the targets is not unique, two nodes having the same importance. In fact that situation would eventually lead to a factorial number of equivalent configurations, failing to exploit the ordering in the first place, and to successfully lower the complexity of the problem.

To guide in a principled way the choice of an ordering, further studies would be required. For instance, it may be of interest to study the impact of the ordering on the selected controllable configuration of controllable nodes in canonical graph models such as Erdös-Rényi random graphs [START_REF] Erdös | On the evolution of random graphs[END_REF] and Barabási-Albert scale free networks [START_REF] Barabási | Emergence of Scaling in Random Networks[END_REF].

Main advantages

Existing methods in general do not allow for target controllability [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Tang | Colloquium: Control of dynamics in brain networks[END_REF], require multiple inputs [START_REF] Klickstein | Energy scaling of targeted optimal control of complex networks[END_REF], or provide approximate solutions [START_REF] Czeizler | Structural Target Controllability of Linear Networks[END_REF][START_REF] Gao | Target control of complex networks[END_REF]. Here, we propose an exact method, able to identify the highest-ranking configuration of targets controllable by a single driver node in a reduced amount of time. This method is highly versatile and can be adapted to different applications.

Biological applications

We exploited the step-wise target controllability framework to model two biological processes: the activation of macrophage cells, and aging in the brain.

Aided by experts, we chose context-specific target sets, and their meaningful ordering. Then, we tested each node as a driver, and we obtained a list of their highest-ranked controllable targets.

Differently from most of the literature, we imposed a single driver node, thus we did not require to achieve control over all of the designated targets. Our work is original in the fact that -once found the controllable configurations -it focuses on the study of functional interactions between controllable pairs driver-target, with the aim to uncover relevant biological relations that could be missed by topological analyses.

Chapter 5

Step-wise target control centrality: biological applications

In this chapter are reported text and figures of two recently submitted papers. The papers discuss the two applications in which we tested our method: the macrophage activation process and aging in the brain. Both applications are studied from the perspective of step-wise target controllability, hence there may be some redundancy in the methodological sections.

Chapter 5.

Step-wise target control centrality: biological applications 28

Macrophage activation network in multiple sclerosis

Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis.

Giulia Bassignana

Abstract

Identifying the nodes able to drive the state of a network is crucial to understand, and eventually control, biological systems. Despite recent advances, such identification remains difficult because of the huge number of equivalent controllable configurations, even in relatively simple networks. Based on the evidence that in many applications it is essential to test the ability of individual nodes to control a specific target subset, we develop a fast and principled method to identify controllable driver-target configurations in sparse and directed networks. We demonstrate our approach on simulated networks and experimental gene networks to to characterize macrophage dysregulation in human subjects with multiple sclerosis.

Introduction

For many biological systems, it is crucial to identify the units, such as genes or neurons, with the potential to influence the rest of the network, as this identification can enable describing, understanding, and eventually controlling the function of the system [START_REF] Bonifazi | GABAergic Hub Neurons Orchestrate Synchrony in Developing Hippocampal Networks[END_REF][START_REF] Jeong | Lethality and centrality in protein networks[END_REF]. Topological descriptors based on network science can indeed be used to quantify such influence in terms of node centrality, such as degree, betweenness, or closeness [START_REF] Newman | Networks: An Introduction[END_REF]. However, these descriptors only capture the structural properties of the network and neglect their effect on the dynamics, thus limiting our understanding on the actual influencing power.

Controllability network theory, linking network structure to dynamics through linear or nonlinear models, has been shown to be a more principled approach for identifying driver nodes in an interconnected system [START_REF] Rugh | Linear System Theory[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. While theoretically these approaches can give a minimum set of driver nodes sufficient to steer the system into desired states, their exhaustive identification might be difficult in practice as there exists in general a very large number of equivalent controllable walks, even in Chapter 5.

Step-wise target control centrality: biological applications 29 relatively simple networks [START_REF] Heuberger | The number of maximum matchings in a tree[END_REF][START_REF] Wagner | On the number of matchings of a tree[END_REF]. In the case of criteria based on the manipulation of controllability matrices [START_REF] Hautus | Stabilization controllability and observability of linear autonomous systems[END_REF][START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF], the presence of many walks can for example induce numerical errors due to the different orders of magnitude in the matrix elements.

An alternative solution has recently been proposed to circumvent this limitation, based on the possibility to map the controllability problem onto the maximum cardinality matching over the associated graph [START_REF] Lin | Structural controllability[END_REF][START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Shields | Structural controllability of multiinput linear systems[END_REF]. As a result, it is possible to identify a set of driver nodes -at least for directed networks -with linear, and not exponential, time complexity [START_REF] Hopcroft | A n5/2 algorithm for maximum matchings in bipartite[END_REF]. While this approach elegantly solves numerical issues, it can nevertheless not tell which configuration, among all the possible ones, is the most relevant. In general, there is a factorial number of equivalent configurations (with the same number of inputs) and enumerating all possible matchings [START_REF] Uno | Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs[END_REF] rapidly becomes unfeasible, even for simple graphs such as trees [START_REF] Heuberger | The number of maximum matchings in a tree[END_REF][START_REF] Wagner | On the number of matchings of a tree[END_REF], bipartite graphs [START_REF] Liu | Number of maximum matchings of bipartite graphs with positive surplus[END_REF], or random graphs [START_REF] Zdeborová | The number of matchings in random graphs[END_REF]. Thus, the research of alternative strategies to characterize the candidate driver nodes is crucial for the concrete application of network controllability tools.

One possibility would be to reduce the original problem into smaller sub-problems under the assumption of specific constraints compatible with the underlying scientific question. On the one hand, for many biological, technological and social systems it is desirable to only control a subset of target nodes (or a subsystem) that is essential for the system mission pertaining to a selected task or function. In this direction some approaches have been recently proposed, based on filtering of the controllability matrix [START_REF] Klickstein | Control Distance and Energy Scaling of Complex Networks[END_REF] or adaptation of graph matching [START_REF] Czeizler | Target Controllability of Linear Networks[END_REF][START_REF] Gao | Target control of complex networks[END_REF][START_REF] Li | Structural Target Controllability of Undirected Networks[END_REF][START_REF] Zhang | Efficient target control of complex networks based on preferential matching[END_REF]. However, they do not solve the problem of multiple driver set configurations. On the other hand, technical and experimental constraints often limit the possibility to stimulate many driver nodes in parallel, for example in gene expression modulation [START_REF] Lodish | Gene Replacement and Transgenic Animals[END_REF] or brain stimulation [START_REF] Hallett | Transcranial magnetic stimulation and the human brain[END_REF]. In these cases, approaches that focus on the ability of single driver node to control the entire network, such as control centrality [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF] or single-node controllability [START_REF] Gu | Controllability of structural brain networks[END_REF], do circumvent the multiplicity issue, but can still suffer from numerical errors and approximate results.

To overcome this impasse, we propose an integrated method that combines the advantages of the previous approaches and quantifies the capacity of a single driver node to control a predefined target set. Based on the Kalman controllability condition, our method identifies the part of the target set that can be controlled by a candidate driver. To do so, we introduce a ranking among the target nodes and we iteratively evaluate the controllability of the system by adding one target node at a time in a descending order. This eventually finds a univocal controllable configuration corresponding to the highest ranking. In the following, we first illustrate how our method, named step-wise target controllability, works for simple network structures Chapter 5.

Step-wise target control centrality: biological applications 30

and we discuss the potential benefits for directed and sparse networks, in terms of space and computational complexity, as compared to alternative approaches. Then, we use it to study molecular networks of macrophage pro-inflammatory activation, derived from ontology-based reconstructions, and identify the driver-target pathway alterations using gene expression data from blood samples of patients affected by multiple sclerosis (MS) and a matched group of healthy controls (HC).

Results

Step-wise target controllability identifies a controllable subset of targets

Let G be a directed graph (or network) of N nodes (or vertices) and L links (or edges), and T an arbitrary subset of S < N nodes in the network. The aim is to measure the ability of each node to drive the state of the target set T from a dynamical system perspective [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. In the case of linear time-invariant dynamics, the number of controllable target nodes can be obtained computing the rank of the target controllability matrix [START_REF] Commault | Functional target controllability of networks: Structural properties and efficient algorithms[END_REF][START_REF] Gao | Target control of complex networks[END_REF][START_REF] Murota | Note on a graph-theoretic criterion for structural output controllability[END_REF]:

Q T = CB CAB CA 2 B • • • CA N -1 B (5.1)
where A is the adjacency matrix of the network, B is a vector identifying the driver node and C is a matrix selecting the rows of A corresponding to the target, or output, nodes (Methods).

A walk in the network consists of an alternating sequence of vertices and edges.

The (i, j) entry of Q T indicates how many walks of length j -1 connect the driver to the target i [START_REF] Biggs | Algebraic Graph Theory[END_REF]. Trivially, all the nodes not traversed by these walks do not contribute to the walk lengths and they can be neglected for the purpose of controllability. By removing the irrelevant nodes from the network, A becomes smaller and this results in a target controllability matrix with less columns. Put differently, we avoid the computation of matrix exponentials corresponding to nonexisting driver-target walks (Methods). In practice, this can be of great advantage for reducing the occurrence of round-off errors during the matrix rank calculations.

For example, this is the case for sparse and directed networks, where fewer nodes are reachable as compared to dense and undirected networks.

This can be easily appreciated in the following example. Let us consider a directed full binary tree with h = 6 levels, with the root node as the candidate driver. Without loss of generality, we randomly position a target in each level and we rank them according to their height in the tree. Then, we introduce a simple cycle among the first three nodes of the tree (Figure 4.1). By construction, this configuration is controllable and the entire target set can be fully driven by the Chapter 5. Step-wise target control centrality: biological applications 31 driver. However, when considering the entire network the returned rank is deficient.

Instead, by removing the part of the network that is irrelevant for the controllability, the rank is full and we retrieve the entire controllable configuration, even in the case of larger networks, i.e. up to h = 10 levels.

The rank of Q T gives the number of target nodes τ ≤ S controllable by the driver, but there might be in general many possible equivalent configurations. To overcome this issue, we propose a step-wise procedure that tests the controllability on subproblems of increasing size. First, we introduce a hierarchy among the target nodes and relabel them according to their importance in a descending order, i.e.

t 1 t 2 ... t S .
Then, we create an empty auxiliary set T and we sequentially include the target nodes according to their ordering. At each step, if the rank of Q T is full, the new target node is retained, otherwise it is removed from T . When all the target nodes have been visited, the algorithm returns the set of controllable targets with highest ranking (Figure 5.1, Methods).

Our method, named step-wise target controllability, not only returns for each candidate driver the number of controllable target nodes τ corresponding to the configuration with highest ranking, but also the set T of controllable targets.

Driver genes are homogeneously distributed in the macrophage network

To test our method in a biological context, we construct a network representing the interactions between molecules involved in macrophages response to proinflammatory stimuli (Figure 5.2), with the connections between genes inferred from a previously established network based on literature [START_REF] Raza | A logic-based diagram of signalling pathways central to macrophage activation[END_REF][START_REF] Robert | Macrophages.com: An on-line community resource for innate immunity research[END_REF]. This network is of interest in MS due to the chronic inflammation characteristic of the disease, and the generally destructive effects of pro-inflammatory macrophages in MS [START_REF] Bitsch | Tumour necrosis factor alpha mRNA expression in early multiple sclerosis lesions: Correlation with demyelinating activity and oligodendrocyte pathology[END_REF].

Hence, dysregulation of macrophages may lead to aggravated inflammation and disease. In order to facilitate biological interpretation of the network, we divide the nodes according to molecular function: sensing, signaling, transcription factors or secreted molecules (Table SA1). We choose the 13 secreted molecules as target nodes because they represent the end-products of macrophage pro-inflammatory activation and enable propagation of inflammation to other cells, thus exacerbating chronic inflammation.

To establish a hierarchy among the targets, we use macrophage RNA expression data from a group of MS patients and healthy controls. The macrophages were tested with and without activating stimuli to mimic the pro-inflammatory response.

We measure the gene activation as the ratio of the expression between the "proinflammatory" and "alert" condition. We then consider the fold change ∆ between The Kalman condition informs us that only two targets are controllable from the driver, i.e. τ = rank(Q T ) = 2. However, there might be up to 3 equivalent configurations that are controllable, i.e. {t 1 , t 2 }, {t 1 , t 3 }, and {t 2 , t 3 }. For larger networks, the number of Kalman tests to perform can be prohibitive, i.e. ( S τ ). Panel b). By introducing a hierarchy among the target nodes, our step-wise method identifies the configuration with the most important nodes by performing only S tests (see Methods). In this example, the first step considers the subgraph containing all the walks from the driver to the target set T = {t 1 }. The associated controllability matrix has full rank, i.e. rank(Q T ) = 1. The first target is therefore retained and the algorithm moves to Step 2, by constructing a new subgraph containing the walks from the driver to the target set T = {t 1 , t 2 }. The rank of the new controllability matrix is now deficient and t 2 is not retained. In Step 3, the new subgraph contains the walks from the driver to T = {t 1 , t 3 }. Because rank(Q T ) is full and there are no more targets, the algorithm stops and returns the controllable configuration t 1 , t 3 .

the gene activation of MS patients and HC subjects (Methods). Genes with larger ∆ values are ranked first (Figure SA1).

Results show that 51% of the tested network nodes can control at least one target (i.e. τ > 0) and that those drivers tend to be homogeneously distributed [START_REF] Raza | A logic-based diagram of signalling pathways central to macrophage activation[END_REF][START_REF] Robert | Macrophages.com: An on-line community resource for innate immunity research[END_REF]. The network consists of N = 101 nodes corresponding to genes involved in inflammation; for the sake of interpretation, they are organized in four classes, depending on their function in the cell. Sensing genes are in the membrane of the cell and start a signaling pathway inside the cell, to the transcription factors, which promote the production of secreted molecules. There are L = 211 directed edges representing either activation or inhibition interactions between molecules (Methods). The size of the nodes is proportional to their total degree k. Panel b) shows gene activation computed as the ratio in expression between the "pro-inflammatory" and "alert" states, based on our RNA sequencing data, generated from monocyte-derived macrophages from blood samples of multiple sclerosis patients (n = 8) and healthy controls (n = 8) (Methods). Solid lines represent group-averaged values, while transparent patches stand for standard deviation. across classes (Tables SA2 andSA3). This indicates a high redundancy in the way the target set can be controlled. Notably, target centrality values are weakly correlated (Spearman rho 0.18, p < 0.07) with the corresponding total node degree k, as defined in Methods, indicating that the most connected genes (e.g. RELA, NFKB1) are not necessarily the ones that can most efficiently steer the state of the Almost all of the driver nodes identified by our method can control the target genes CCL5, CXCL10, CXCL11, IFNA1 and IFNB1, which code for inflammatory chemokines and cytokines. This implies a high level of co-regulation among these molecules, with many different actors exerting control over this regulation. Interestingly, the drivers with the highest step-wise target control centrality (τ ) values (SOCS1 and SOCS3, τ = 10) belong to feedback systems that control pro-and anti-inflammatory signal transduction by regulating the signaling process triggered in response to IFNγ [START_REF] Mccormick | Regulation of Macrophage, Dendritic Cell, and Microglial Phenotype and Function by the SOCS Proteins[END_REF]. In addition, all drivers with τ 9 can be seen in our network as a cluster of genes converging onto and including STAT1 (Figure 5.3a).

This cluster includes the receptors of IFNγ and the signaling molecules responsible for their intracellular effects. This result matches the well-described effects of IFNγ on chemokine production [START_REF] Koper | CXCL9, CXCL10, CXCL11, and their receptor (CXCR3) in neuroinflammation and neurodegeneration[END_REF][START_REF] Liu | Interferon regulatory factor 1 is an essential and direct transcriptional activator for interferon {gamma}-induced RANTES/CCl5 expression in macrophages[END_REF] and overall macrophage activation [START_REF] Mosser | Exploring the full spectrum of macrophage activation[END_REF].

Robustness of driver nodes to random attacks

To assess the stability of our findings to possible errors in the network construction, we performed a robustness analysis simulating different types of alterations to its nodes and links (Methods).

Results show that removing nodes with higher degree k leads to a greater reduction of step-wise target control centrality in the drivers compared to the removal of lowdegree nodes or random removal of nodes (Figure 5.3b). For example, by attacking 10% of the nodes we lose 5% of the drivers in the latter cases, while we lose 20% of the drivers when removing the most connected ones (Figure 5.3b). This result confirms the crucial role of hubs in biological networks in terms of resilience to random attacks [START_REF] Barabási | Network biology: Understanding the cell's functional organization[END_REF] and controllability [START_REF] Pu | Robustness analysis of network controllability[END_REF].

When perturbing links, the worst condition is given by their random removal.

By attacking 10% of the links around 5% of the drivers are lost. This is intuitively due to the interruption of driver-target walks and to consequent impossibility to control a node that cannot be reached. While randomly rewiring the links has an intermediate impact, adding new links has no effect on the step-wise target control centrality of the drivers (Figure 5.3c). This is of great advantage as it shows that our results will not change if new connections are established or provided by the literature.

Gene dysregulation and altered driver-target coactivation in multiple sclerosis

Using step-wise target controllability, we detect potential directed interactions in the macrophage activation network, but we cannot quantify how changes in the driver's state affect those in the targets. To measure driver-target functional interactions, we compute the Spearman correlation between the gene activation of controllable drivertarget pairs, for the HC and MS groups (Methods, Tables SA4 to SA7). We call coactivated the genes exhibiting a significant correlation (p< 0.05). Results show that in general only a moderate fraction (21%) of all the possible driver-target genes are coactivated (Figure 5.4a, Tables SA4 to SA7). For both HC and MS groups these interactions tend to primarily involve signaling functions (Figure 5.4b). However, the number of driver-target coactivations is lower in the pathological condition (M S = 19 versus HC = 36). More importantly, they differ from those observed in the HC group (Figure 5.4b). This is particularly evident for target IFNA1, which Because the macrophage network edges are fixed and reconstructed from known protein-protein interactions, differences in coactivation can be essentially attributed to altered regulation of transcription. Hence, our hypothesis is that the observed functional reorganization can be explained by the dysregulation of specific genes along the controllable walks from the drivers to targets. To test this prediction, we examine all the pairs of genes whose coactivation appears or disappears in the MS group (Figure 5.4a). We found that 47/51 of these differentially coactivated pairs present at least one dysregulated gene (i.e., fold change |∆| above the 75-percentile, Methods) on the walk from the driver to the target (Figure 5.5, Tables SA8 to SA12).

We find in total 14 dysregulated nodes on any of these walks. The genes that most frequently appear are NFKB1, IFNA1 and IFNB1 (36/51 walks). They are present on all walks that end with targets CCL5, CXCL10, CXCL11, IFNA1 or IFNB1, i.e., the 5 targets that could be controlled by most drivers. This points to their dysregulation being a potent disruptor of the normal network functioning. The co-occurrence of these three dysregulated genes can be explained by a feedback loop in which NFKB1 activates IFNB1, and IFNA1 and IFBN1 both activate STAT2, which through several intermediates can influence all three genes (Figure SA2).

Indeed, this stems from the fact that all these nodes belong to the main connected component of the network, i.e. a subnetwork in which every node is reachable from any other node.

Taken together, these results indicate that the aberrant reorganization of functional interactions in the MS group is associated with the presence of dysregulated genes along the controllable walks of the macrophage network.

Switch of SOCS-gene coactive drivers reflects dysregulated inflammatory

response Because drivers are crucial for steering the target network's state, we focus on the subnetwork specifically involving the dysregulated drivers (IRF8, NFKB1, SOCS1, SOCS3, TLR7) and the walks towards the respective controllable targets est τ centrality values, i.e. SOCS1 and SOCS3. In the HC group, SOCS1 is coactivated with the targets while SOCS3 does not exhibit any significant correlation. In the MS group, we observe the opposite, i.e. SOCS1 is silent while SOCS3 becomes coactive. Because both driver genes are dysregulated, the observed "switch" mechanism could be therefore associated with the altered pro-inflammatory response of the MS group. Indeed, these two molecules are known to be strong modulators of macrophage response: SOCS1 inhibits the signaling of pro-inflammatory genes while SOCS3 is known to be an important actor in inflammatory response, with the ratio of the two proteins determining the actual effect [START_REF] Wilson | SOCS Proteins in Macrophage Polarization and Function[END_REF]. Chapter 5.

Step-wise target control centrality: biological applications 40

Discussion

Identification of controllable configurations in complex networks

Network controllability refers to the ability to drive an interconnected dynamical system from any initial state to any desired final state in finite time, through a suitable selection of inputs [START_REF] Rugh | Linear System Theory[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. In recent years, an increasing number of research groups from different disciplines have focused their efforts on identifying the minimum set of driver nodes or quantifying the capacity of single nodes to control the entire network, as well as parts of it [START_REF] Betzel | Optimally controlling the human connectome: The role of network topology[END_REF][START_REF] Dosenbach | Distinct brain networks for adaptive and stable task control in humans[END_REF][START_REF] Gu | Optimal trajectories of brain state transitions[END_REF][START_REF] Liu | Control Principles of Complex Networks[END_REF][START_REF] Lugagne | Balancing a genetic toggle switch by real-time feedback control and periodic forcing[END_REF][START_REF] Menara | On Structural Controllability of Symmetric (Brain) Networks[END_REF][START_REF] Muldoon | Stimulation-Based Control of Dynamic Brain Networks[END_REF][START_REF] Sun | Controllability and stability analysis of large transcriptomic dynamic systems for host response to influenza infection in human[END_REF][START_REF] Tang | Developmental increases in white matter network controllability support a growing diversity of brain dynamics[END_REF][START_REF] Wagner | Noninvasive Human Brain Stimulation[END_REF][START_REF] Wuchty | Controllability in protein interaction networks[END_REF]. Despite being theoretically attractive, network controllability still suffers from computational issues that limit its impact in concrete applications. This is mainly due to the presence of multiple equivalent controllable walks in a network that make the associated controllability problem ill-posed and/or the resulting solution space very big [START_REF] Rugh | Linear System Theory[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF].

To reduce such complexity, we propose a method based on control centrality, which was previously designed to quantify the ability of one node to control directed networks [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF]. First, we define the target control centrality to measure the controllability of a specific part of the network, i.e., a predefined target set. Because edges are directed, this has the advantage to ignore the part of the network that is not traversed by the walks connecting the driver to the target set. Second, we introduce an ordering among the target nodes and perform a step-wise controllability test with increasing size. Because of the ranking, only one controllable configuration will be identified, i.e. the one with the highest ranking (Figure 5.1). To test the controllability of the driver-targets configuration at each step we adopted the Kalman criterion [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Rugh | Linear System Theory[END_REF]. However, the entire iterative framework is quite flexible and other methods, such as Gramian condition [START_REF] Klickstein | Energy scaling of targeted optimal control of complex networks[END_REF], Popov-Belevitch-Hautus criterion [START_REF] Li | Structural Target Controllability of Undirected Networks[END_REF], or feedback vertex set [START_REF] Zañudo | Structure-based control of complex networks with nonlinear dynamics[END_REF], could be used as alternative controllability criteria.

While the step-wise target controllability achieves the identification of one solution in a significantly reduced amount of time, it is important to state that the method for ordering of the targets is a subjective choice. For example, targets can be sorted according to their importance in a biological function (e.g. genes) [START_REF] Liseron-Monfils | NECorr, a Tool to Rank Gene Importance in Biological Processes using Molecular Networks and Transcriptome Data[END_REF][START_REF] Zhao | Ranking Candidate Disease Genes from Gene Expression and Protein Interaction: A Katz-Centrality Based Approach[END_REF] or in discriminating pathological conditions (e.g. brain areas) [START_REF] Lohmann | Eigenvector Centrality Mapping for Analyzing Connectivity Patterns in fMRI Data of the Human Brain[END_REF][START_REF] Sen | Ranking Regions, Edges and Classifying Tasks in Functional Brain Graphs by Sub-Graph Entropy[END_REF].

The choice of sorting criteria should reflect the specific scientific question. In a more network-centric approach, ordering could rely on the ranking of topological network properties such as node centrality measures [START_REF] Newman | Networks: An Introduction[END_REF]. While this is beyond the scope of our study, we feel that this should be more extensively investigated by future research.

Chapter 5. Step-wise target control centrality: biological applications 41

Control pathways in macrophage molecular networks

The study of the molecular interactions is crucial to the understanding of the basic functions of the cell such as proliferation or apoptosis [START_REF] Maslov | Specificity and Stability in Topology of Protein Networks[END_REF][START_REF] Taylor | Dynamic modularity in protein interaction networks predicts breast cancer outcome[END_REF]. Determining the connection mechanisms that rule a specific biological function can significantly impact our daily life by providing new therapeutics to counteract diseases [START_REF] Drier | Pathway-based personalized analysis of cancer[END_REF][START_REF] Menche | Disease networks. Uncovering disease-disease relationships through the incomplete interactome[END_REF][START_REF] Tong | A novel grading biomarker for the prediction of conversion from mild cognitive impairment to Alzheimer's disease[END_REF].

Studying molecular networks is however difficult, because in general we do not know the true functional interactions of a cell and indirect techniques such as gene coexpression are typically employed to infer such connections [START_REF] Uygun | Utility and Limitations of Using Gene Expression Data to Identify Functional Associations[END_REF]. Based on correlation analysis, these methods cannot inform on the causal nature of the interactions.

More importantly, the reliability of the estimated network critically depends on the number of interactions to number of data samples ratio, which is in practice very low [START_REF] Uygun | Utility and Limitations of Using Gene Expression Data to Identify Functional Associations[END_REF].

To overcome this limitation, we reconstruct the directed gene interactions associated with the inflammatory state of the human macrophages by adopting a novel ontology-based approach that integrates the available information from multiple datasets and results in the literature [START_REF] Henry | BiPOm: Biological interlocked Process Ontology for metabolism. How to infer molecule knowledge from biological process?[END_REF]. Previous studies show that the number of driver nodes in biological networks is rather high due to their sparse and heterogeneous nature [START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Ruths | Control Profiles of Complex Networks[END_REF]. Consistently, we find that a large percentage of genes (51%) can control at least one secreted molecule in the target set. Our results also confirm that, despite being crucial for global communication, hubs (e.g. RELA) are not always the most important from a network controllability perspective (Figure 5.3a).

This stems from the theoretical impossibility to diversify the input signals to all the connected neighbors [START_REF] Liu | Controllability of complex networks[END_REF]. The found driver genes are heterogeneously distributed across the tested gene classes. However, our method highlights SOCS1 and SOCS3

as the drivers with the highest step-wise target control centrality, with other IFNγresponse-related genes showing similar values. This is in line with the known effects of SOCS-genes and IFNγ on molecules secreted by pro-inflammatory macrophages [START_REF] Koper | CXCL9, CXCL10, CXCL11, and their receptor (CXCR3) in neuroinflammation and neurodegeneration[END_REF][START_REF] Liu | Interferon regulatory factor 1 is an essential and direct transcriptional activator for interferon {gamma}-induced RANTES/CCl5 expression in macrophages[END_REF][START_REF] Mccormick | Regulation of Macrophage, Dendritic Cell, and Microglial Phenotype and Function by the SOCS Proteins[END_REF][START_REF] Mosser | Exploring the full spectrum of macrophage activation[END_REF], supporting the ability of this method to identify biologically relevant drivers.

Overall, these results uncover the existence of potential causal influences from candidate driver genes to the secreted molecules in the human macrophage activation network. Because the identified driver nodes are robust to network alterations, notably when adding new links (Figure 5.3c), the obtained results are expected to be sufficiently resilient to the integration of new gene-gene interactions. From a different angle, our approach can be seen as a new way to filter information in complex networks and focus on the specific nodes (the drivers) or node pairs (driver-target).

This might have important consequences when studying genome-wide databases where the high number of elements can make prohibitive the assessment of signifi-Chapter 5.

Step-wise target control centrality: biological applications 42 cant gene expressions and/or co-expressions [START_REF] Bentley | [END_REF][START_REF] Mullighan | Genome-wide analysis of genetic alterations in acute lymphoblastic leukaemia[END_REF].

Dysregulated genes and aberrant interactions in multiple sclerosis

Multiple sclerosis is an immune-mediated disease in which the immune system erroneously attacks myelin in the central nervous system. There are many neurological symptoms, including motor and cognitive deficits, that can vary in type and severity depending on the attacked central nervous system regions [START_REF] Hauser | Multiple Sclerosis[END_REF]. The role of macrophages in MS is crucial because of their ability to obtain a pro-inflammatory activation state, including the release of pro-inflammatory cytokines and leading to central nervous system tissue damage [START_REF] Chu | The roles of macrophages and microglia in multiple sclerosis and experimental autoimmune encephalomyelitis[END_REF]. Hence, dysregulation of macrophages may lead to autoimmunity and persistent inflammatory diseases [START_REF] Strauss | The immunophenotype of antigen presenting cells of the mononuclear phagocyte system in normal human liver -A systematic review[END_REF]. While the etiology of MS is still not well-understood there is a large consensus on its genetic basis and on the importance of unveiling the underlying network mechanisms [START_REF] Airas | Hormonal and gender-related immune changes in multiple sclerosis[END_REF].

In this study, we combined network controllability tools and gene expression data to detect the genes responsible for altering the macrophages action in multiple sclerosis. Differently from standard approaches, where the attention is focused on the identification of the driver nodes in a network, we here propose an alternative way of exploiting network controllability. We first show that the macrophage inflammatory state in the MS group was characterized by a drastic alteration of the coactivations in the driver and target genes (Figure 5.4). Such absence of coordination was in general associated with the presence of dysregulated genes along the walks from the driver to the target node. Notably, the pathological dysregulation of NFKB1, IFNB1

and IFNA1, which belong to the same feedback cycle (Figure SA2), critically affects several driver-target functional interactions (Figure 5.5).

Finally, our approach allows to identify a shift mechanism for dysregulated SOCS1 and SOCS3 drivers, showing opposite coactivation patterns in MS patients compared to the healthy controls (Figure 5.6). These results suggest that experimentally stimulating SOCS3 -a strong inducer of pro-inflammatory response -might be more effective for moving the state of the altered secreted molecules towards physiological configurations. Taken together, these results might have practical consequences on how to design intervention strategies and counteract disease phenotype.

Methodological considerations

Our method uses Kalman controllability rank condition [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF] to quantify the centrality of the driver nodes. This criterion assumes that the investigated system has a linear dynamics, Equation 5.2. In our case, this means that the changes in the Chapter 5.

Step-wise target control centrality: biological applications 43 gene activation would follow a linear trend. While this is in general not true and difficult to ascertain, it appears that results from non-linear tests are often dominated by linear relationships [START_REF] Song | Comparison of co-expression measures: Mutual information, correlation, and model based indices[END_REF][START_REF] Steuer | The mutual information: Detecting and evaluating dependencies between variables[END_REF]. Furthermore, a significant fraction of the data analysis and modeling deals exclusively with linear approaches as they are simpler, easy to interpret and serve as a prerequisite of nonlinear behavior [START_REF] Liu | Control Principles of Complex Networks[END_REF].

Another peculiarity of our approach is the assumption of time-invariant interactions in the molecular gene network. On the one hand, this assumption allows to better exploit the well-established results and tools in network controllability [START_REF] Liu | Controllability of complex networks[END_REF];

on the other hand, it might conflict with existing literature looking for biological connectivity changes between conditions or populations such as differential gene coexpression [START_REF] Choi | Differential coexpression analysis using microarray data and its application to human cancer[END_REF]. Here, we hypothesized that the activation state of each node (in terms of gene expression) could eventually change but not the underlying network structure. Thus, our network -obtained from detailed maps of the macrophage cells -would only act as a substrate/proxy for functional interactions, such as correlated gene activities.

The implementation of our algorithm does not account for the case in which the ordering of the targets is not unique, two nodes having the same importance. One possible solution may be to add equally important targets at the same iteration step (that is together instead of one at a time) and test if they can be controlled together.

This may not be ideal, because the algorithm will consider them controllable only if both of them can be controlled, still it avoids the combinatorial problems linked with testing multiple targets. However, in many biological contexts such as the one we used, the ordering is based on continuous values and will in general be unique.

We finally notice that our method is conceived for directed networks only, where the dimensionality reduction has a real computational benefit. In fact, in the case of undirected graphs, it is not possible to remove nodes on the walks from the driver to the targets since information is bound to span the entire network. Similarly, for directed but dense networks, the possibility to focus on specific parts of the network, and reduce the computational cost, becomes lower regardless of the topology.

To conclude, it is important to mention that extensions of network controllability tools to time-varying frameworks do exist [START_REF] Li | The fundamental advantages of temporal networks[END_REF][START_REF] Zhang | Controllability of temporal networks: An analysis using higher-order networks[END_REF]. However, in that case networks would be inferred from gene coexpression and therefore affected by statistical uncertainty due to sample sizes. Further research is needed to seek how to apply network controllability in presence of noisy time-varying connections.

Conclusion

In this study, we introduce a method to quantify the ability of candidate driver nodes to drive the state of a target set within a sparse and directed network. Further,

Chapter 5. Step-wise target control centrality: biological applications 44

we illustrate how this method works for the molecular network associated with the human macrophage inflammatory response. The obtained results reveal in a principled way the genes that are significantly dysregulated in multiple sclerosis.

We hope that this method can contribute to the identification of the key nodes in biological networks to better identify pharmacological targets to counteract human diseases.

Methods

Step-wise target controllability

We introduce a method to identify which target nodes in a network that can be controlled from a single driver node. To do so, we start by considering the canonical linear time-invariant dynamics on a directed network described by the adjacency

matrix A ∈ R N ×N ẋ(n) = Ax(n) + Bu(n), y(n) = Cx(n) (5.2)
where x(n) ∈ R N describes the state of each node at time n, B ∈ R N specifies the driver node, u(n) ∈ R N is its external input (or control) signal, y(n) ∈ R S is the output vector, and C ∈ R S×N is the output matrix identifying the target nodes.

Such a system is controllable if it can be guided from any initial state to any desired final state in finite time, with a suitable choice of input. A necessary and sufficient condition to assess the controllability of Equation 5.2, is that the con-

trollability matrix Q Q = B AB A 2 B • • • A N -1 B (5.3)
has full row rank, i.e. rank(Q) = N . That is the Kalman rank condition, which basically verifies the existence of linearly independent rows in Q [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Rugh | Linear System Theory[END_REF]. If so, the driver node can reach and control the dynamics of all the other nodes through independent walks of length N -1 at maximum.

If it is of interest to control only a target set T of the network, specified in C and consisting of S ≤ N nodes, then Equation 5.2 can be reduced into a target controllability matrix Q T = CQ (Equation 5.1), where C filters the rows of interest corresponding to the targets. Now, the rank of Q T gives the number τ ≤ S of nodes in the target set that can be controlled by the driver.

To identify a driver-target configuration, we further introduce a hierarchy among the target nodes, so that we can order and relabel them from the most important one to the least, i.e.

-If rank(Q T ) is full then τ ← τ + 1 else T ← T \ {t j } -j ← j + 1 • Step 3. Output τ and T
Eventually, the step-wise target control centrality (τ ) is the number of controllable targets in T , and the set T contains the τ controllable targets with highest ranking.

Construction of the macrophage activation network

We reconstruct the inflammatory molecular network of the human macrophage by integrating information from the macrophage signal transduction map [START_REF] Raza | A logic-based diagram of signalling pathways central to macrophage activation[END_REF][START_REF] Robert | Macrophages.com: An on-line community resource for innate immunity research[END_REF].

This map contains a comprehensive, validated, and annotated map of signal transduction pathways of inflammatory processes in macrophages based on the current literature. To extract molecular interactions from this map, we used the Hermit software [START_REF] Motik | Structured objects in owl: Representation and reasoning[END_REF], which implements automatic reasoning based on logical rules. We specifically used the rules implemented in the molecular network ontology to infer molecular interactions depending on the process they belong [START_REF] Henry | BiPOm: Biological interlocked Process Ontology for metabolism. How to infer molecule knowledge from biological process?[END_REF][START_REF] Musen | The Protégé Project: A Look Back and a Look Forward[END_REF]. Because we are interested in the inflammation process, we restricted our analysis to a specific subset of 101 genes with known roles in macrophage pro-inflammatory activation, and for which their regulation in response to pro-inflammatory stimuli could be confirmed in our data set. These genes were classified according to their function in the cell: sensing, signaling, transcription and secreted (Table SA1), as described in databases such as NCBI Gene [START_REF] Agarwala | Database resources of the National Center for Biotechnology Information[END_REF], UniProt [START_REF] Consortium | UniProt: A worldwide hub of protein knowledge[END_REF] and GeneCards [START_REF] Stelzer | The GeneCards Suite: From Gene Data Mining to Disease Genome Sequence Analyses[END_REF]. The full network was thus reduced to only include these genes and their interactions. Due

Chapter 5. Step-wise target control centrality: biological applications 46

to recent studies, we also opted to exclude two edges (from SOCS3 to IFNGR1 and to IFNGR2) to represent the involved pathways [START_REF] Wilson | SOCS Proteins in Macrophage Polarization and Function[END_REF].

The resulting network contains N = 101 nodes and L = 211 unweighted directed edges representing either activation or inhibition between genes. The total degree k of each node in the network is computed by summing the number of incoming and outgoing edges:

k i = N j=1 A ij + N j=1
A ji (5.4) where A ij = 1 if there is an edge between the corresponding genes, and 0 otherwise.

Collection of macrophage mRNA expression data

Collection of blood for the study was approved by the French Ethics committee and the French ministry of research (DC-2012-1535 and AC-2012-1536). Written informed consent was obtained from all study participants. All patients fulfilled diagnostic criteria for multiple sclerosis [START_REF] Thompson | Diagnosis of multiple sclerosis: 2017 revisions of the McDonald criteria[END_REF], and individuals (multiple sclerosis patients and healthy donors) with any other inflammatory or neurological disorders were excluded from the study. Patients were included in the study only if they were not undergoing treatment.

Blood was sampled from 8 MS patients and 8 healthy controls in acid citrate dextrose tubes. From blood samples, peripheral blood mononuclear cells were isolated using Ficoll Paque Plus (www.gelifesciences.com) and centrifugation (2200 rpm, 20 min). Cells were washed in PBS and RPMI +10% FCS. Monocytes were isolated with anti-CD14 microbeads (www.miltenyibiotec.com) and plated in 12well plates (500000 cells/well) in RPMI +10% FCS and granulocyte-macrophage colony-stimulating factor (500 U/ml) to induce differentiation into macrophages.

After 72h, media was replaced with fresh media supplemented with granulocytemacrophage colony-stimulating factor (500 U/ml) to maintain "alert" macrophages or IFNγ (200 U/ml) + upLPS (10 ng/ml) to induce "pro-inflammatory" activation. Cells were lysed after 24h and RNA was extracted with RNeasy Mini Kit (www.qiagen.com).

Transcriptome sequencing cDNA libraries were prepared using a stranded mRNA polyA selection (Truseq stranded mRNA kit, www.illumina.com). For each sample, we performed 60 million single-end, 75 base reads on a NextSeq 500 sequencer (www.illumina.com). RNA-Seq data analyses were performed by GenoSplice technology (www.genosplice.com). Sequencing, data quality, reads repartition (e.g., for potential ribosomal contamination), and insert size estimation are performed using FastQC [START_REF] Andrews | FastQC: A Quality Control Tool for High Throughput Sequence Data[END_REF], Picard-Tools (http://broadinstitute.github.io/picard/), Samtools Chapter 5.

Step-wise target control centrality: biological applications 47 [START_REF] Li | The Sequence Alignment/Map format and SAMtools[END_REF] and rseqc [START_REF] Wang | RSeQC: Quality control of RNA-seq experiments[END_REF]. Reads were mapped using STARv2.4.0 [START_REF] Dobin | STAR: Ultrafast universal RNA-seq aligner[END_REF] on the hg19 Human genome assembly. Gene expression regulation study was performed [START_REF] Noli | Discordant Growth of Monozygotic Twins Starts at the Blastocyst Stage: A Case Study[END_REF]. Briefly, for each gene present in the FAST DB v2018_1 annotations, reads aligning on constitutive regions (that are not prone to alternative splicing) were counted. Based on these read counts, normalization was performed using DESeq2 [START_REF] Love | Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2[END_REF] in R (v.3.2.5) [START_REF] Team | R: A Language and Environment for Statistical Computing[END_REF].

Network modeling and data analysis

In the modeling framework described by Equation 5.2, matrix A corresponds to the molecular network and represents the time-invariant component of the system.

The dynamic component is instead represented by the gene activation response in the healthy and diseased condition (Figure 5.2b), computed as the ratio in gene expression between the "pro-inflammatory" and "alert" condition. Specifically, x(n)

represents the gene activation. B is a vector identifying the candidate driver. The control signal u(n) is out of the scope of this work. The output vector y(n) and the output matrix C identify the target nodes.

We select the genes belonging to the secreted molecules class (Table SA1) as our target set T . All the nodes in the other classes are then tested separately as potential driver nodes by computing their step-wise target control centrality (τ ). To enhance numerical precision, the logarithmic transformation log (q + 1) is applied to the elements of the target controllability matrix Q T (Equation 5.1).

The hierarchy among the target nodes is established by computing the fold change ∆ between the corresponding gene activation in the two groups: We perform a robustness analysis to evaluate the stability of the identified driver nodes to potential errors in the molecular network reconstruction. We simulate attacks with increasing intensity, i.e. up to 20% of the nodes or edges in the network.

∆ = µ M S µ HC ( 5
When removing nodes, we consider the following cases: i) random deletion, ii) preferential removal of high-degree nodes, and iii) preferential removal of low-degree Chapter 5.

Step-wise target control centrality: biological applications 48 nodes. Preferential attacks are performed by selecting nodes with a probability p proportional to their degree k, i.e. p ∝ k for high-degree nodes and p ∝ -k for low-degree nodes. When perturbing edges, we test: i) random addition, ii) random deletion, and iii) random rewiring. For each case, we simulated 1000 repetitions and we computed the step-wise target control centrality (τ ) for the driver nodes identified in the original network. Then, we report the percentage of nodes that cease to be drivers (i.e. τ = 0), that is, the percentage of nodes that are drivers in our analysis, but are no longer able to control any target in the perturbed case.

Acknowledgments

We would like to thank Prof. Albert-Laszlo Barabasi for his helpful comments and Step-wise target control centrality: biological applications 49

Aging brain network

Abstract

Identifying the nodes that may be able to impact the state of a network is a crucial question for many complex interconnected systems. In particular for biological systems, it is a relevant question because it is often possible to only act on single driver nodes and desirable to focus on their ability to control a target node-set of the network. In the case of the brain, there has been a growing interest in characterizing the brain areas most relevant for different cognitive processes. Motivated by the need to understand what may be the single brain regions able to affect specific structural systems, we implemented the step-wise target controllability framework in human connectomes of 171 healthy subjects aged from 5 to 85. We used diffusion tensor imaging (DTI) and functional magnetic resonance imaging (fMRI) to derive, respectively, structural and functional brain networks, and we directed the structural connectomes with a biased random walk process. Results showed that the sensorimotor system tends to have higher centrality than the other systems for all target sets. Interestingly, we found that for most brain regions the centrality tends to be negatively correlated with age, suggesting that their driving potential deteriorates with normal aging. Our work is a preliminary step towards a better understanding of the aging process in healthy subjects.

Introduction

Network controllability has recently received a growing interest from the scientific community, in particular for what concerns biological systems [START_REF] Ravindran | Identification of critical regulatory genes in cancer signaling network using controllability analysis[END_REF][START_REF] Ravindran | Network controllability analysis of intracellular signalling reveals viruses are actively controlling molecular systems[END_REF][START_REF] Vinayagam | Controllability analysis of the directed human protein interaction network identifies disease genes and drug targets[END_REF][START_REF] Yan | Network control principles predict neuron function in the Caenorhabditis elegans connectome[END_REF].

Great advantages could be provided by a successful application of controllability, which could achieve the ability to steer a system from an initial state to a desired final state. In particular, controllability of brain network has been studied for the possibility to better understand how the brain is able to modify its dynamics in order to enable different cognitive tasks, or cognitive control, and to exploit it in medical treatments [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Muldoon | Stimulation-Based Control of Dynamic Brain Networks[END_REF][START_REF] Tang | Colloquium: Control of dynamics in brain networks[END_REF].

This kind of works addressed questions on whether the brain could theoretically be controlled, and which brain regions may be the best-suited to influence functional states. Centrality measures based on controllability theory were developed and applied to brain networks in order to gain insight in its functioning [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Pasqualetti | Controllability Metrics, Limitations and Algorithms for Complex Networks[END_REF].

These studies have practical applications and can inform medical treatments. For instance, non-invasive transcranial magnetic stimulation [START_REF] Bikson | Safety of Transcranial Direct Current Stimulation: Evidence Based Update 2016[END_REF] and invasive deep brain stimulation [START_REF] Wilson | Clustered Desynchronization from High-Frequency Deep Brain Stimulation[END_REF] are used to treat Alzheimer's disease, Parkinson's disease, stroke, and epilepsy [START_REF] Tang | Colloquium: Control of dynamics in brain networks[END_REF].

Chapter 5. Step-wise target control centrality: biological applications 50

However, the question of how single brain regions can affect not the whole, but parts, or a subsystem, of the brain is yet to be explored.

In this paper, we aim to study from a control-theoretical perspective the role of single regions of interest (ROIs), detailed in Tables SB1 andSB2, in controlling specific target sets. Targets are chosen to be structural systems of the connectome.

We analyzed frontal, limbic, temporal, sensorimotor, parietal, and occipital systems,

Figure SB1
. We implemented the step-wise target controllability framework [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF],

that assumes a linear time invariant dynamics and provides a centrality measure based on the Kalman rank condition [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Luenberger | Introduction to Dynamic Systems: Theory, Models, and Applications[END_REF]. The step-wise target control centrality counts the number of target nodes that can be controlled by a single driver.

We used a database containing information on both the functional (fMRI) and the structural (DTI) brain networks for 171 subjects, aged from 5 to 85 [START_REF] Brown | Connected brains and minds-The UMCD repository for brain connectivity matrices[END_REF]. To study more efficiently the controllability property, we directed the connectomes through a biased random walk process [START_REF] Gómez-Gardeñes | Entropy rate of diffusion processes on complex networks[END_REF], Methods. We asked how is the centrality measure distributed in the structural systems at the varying of the target set, and how does it change with age. Moreover, we studied what happens in the case of a stroke, or a lesion localized to one hemisphere of the brain.

We found that, independently on the target set, the sensorimotor system presents the highest centrality values, indicating that it may be well suited to control target areas of the brain. On the other hand, the limbic system always presents low centrality values. Interestingly, we found that most brain regions tend to lose importance as time passes. In the case of a lesion, we clearly see that the hemisphere in which the lesion is located loses the most in centrality values. In particular, when the sensorimotor system is attacked global losses are the most severe, instead when damaging the limbic system, global losses in the centrality values are contained.

Our work may be seen as a step forward in the identification of the brain areas that could be most relevant in the aging process, viewed in the original perspective of target controllability.

Results

The shows τ centrality values for each brain region when the limbic system is the target set. This case is also exemplary of the case in which one among the frontal, temporal, parietal, or occipital system is the target set. In these configurations, the sensorimotor system has the largest values of centrality, suggesting that it may be well-suited to control the other systems. On the other hand, for all target configurations the limbic system presents low centrality values. Panel b)

shows τ centrality values for each brain region when the sensorimotor system is the target set. In this case the values are extremely low for all structural systems, suggesting that the sensorimotor system may be difficult to control.

across systems and across subjects in the case of different target systems. Results for all target configurations are reported in Figure SB2. We performed a one-way anova [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition[END_REF] to test for the effect of the system on the centrality values, and we found a significant effect (F-values in Table SB3). Moreover, when the sensorimotor system is the target, all other systems present extremely low τ values, that is, only a small number of nodes in the sensorimotor system can be controlled at the same time by a single brain region in the other structural systems, Figure 5.7b. On the other hand, the limbic system presents low centrality values for all target configurations, hinting at its inability to control other systems.

Two exemplary cases of

Chapter 5. Step-wise target control centrality: biological applications 52

Figure 5.8: Self-regulation score averaged across systems and subjects, at the varying of the target set. Self-regulation for a system is computed as the ratio between average τ in that system and total average τ . Thus, self-regulation above (below) 1 indicates that nodes in that system can control more (less) targets than an average node. Across all configurations apart for the limbic system, the target system itself has the largest values of self-regulation, partially explained by the fact that each target node can at least control itself. In all configurations, the limbic system has low self-regulation. Panels a-f) show how self-regulation scores are distributed on the connectomes at the varying of the target set. Panels g-l) show numerical values of the self-regulation score for the corresponding target sets. Error bars stand for standard error across subjects. The light purple marker denotes the target system.

These results indicate that the sensorimotor system may be well suited control other systems, and at the same time it may be difficult to control.

Self-regulation An additional step in analyzing the power of brain regions in controlling a specific target set is to quantify how much of the centrality is concentrated in that system. It may be of particular interest to study how much can a certain system control itself, that is to understand how much a brain region in the target set can control other ROIs in the same system.

We asked whether the centrality values of the target system are higher than those of the other systems. To quantify it, we computed a self-regulation score, made of the ratio between average τ in a system and average τ in all the network, see Methods.

When this score is below (above) one, it indicates that, on average, brain regions Chapter 5.

Step-wise target control centrality: biological applications 53

in the system under consideration can control a smaller (larger) number of target nodes than an average ROI. We computed self-regulation scores for the six target systems, results are shown in Figure 5.8. Then, we performed a one-way anova [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition[END_REF] to test for the effect of the system on the self-regulation scores, and we found a significant effect (F-values in Table SB3).

We found that in general the target system has a larger self-regulation score than the other systems, meaning that a node in the target system can, on average, control more targets than an average node. This may be partially explained by the fact that, by construction, nodes in the target system can at least control themselves.

An exception to this is given by the limbic system, which always has self-regulation score below one, even when it is the target set. The sensorimotor system consistently presents high values of the self-regulation score, having always values above one.

The step-wise target control centrality of the brain regions deteriorates with age

Next, we asked whether there is a change in the step-wise target control centrality (τ ) while growing old. In particular, we were interested in studying which ROIs present a significant change in step-wise target control centrality as age increases.

We computed the Spearman partial correlation between τ and age for each target system, results are reported in Figure SB3 and Tables SB4 to SB7. We corrected for outdegree in order to disentangle the contribution of outgoing connections from that of the centrality. The outdegree is relevant to the controllability property, because outgoing edges make possible to control other nodes. However, because of the procedure we implemented to direct the brain networks, indegree is almost perfectly negatively correlated with outdegree (Spearman rho = -0.97, p < 10 -6 ), see Methods.

When different target sets are considered, the centrality values change. Thus, the correlation may vary for different target sets. ROIs that are significant (p < 0.05) in at least one target configuration are shown in Figure 5.9a. We focused on those regions that are most consistent across target sets, that is, the regions which consistently present a negative (or positive) correlation across target configurations.

We found that 109 out of 158 regions, 69%, are significant for some target set.

Among those, 82 (75%) present a negative trend, and 24 (22%) a positive one. Only for three regions, 3%, (LLG2, RFP10, RPOC1) the correlation changes in sign when different target configurations are considered.

Six regions are consistent across all six target sets, and present a negative correlation. Among those, the right lateral occipital inferior 3 (RLOCid3) region is in the occipital system, and is known to be part of the dorsal attention network and age, corrected for outdegree. The correction is aimed at disentangling the contribution of outgoing connections and that of the centrality. The outdegree is relevant to the controllability property, in fact outgoing edges make possible to control other nodes. Panel a) shows brain regions that present a significant correlation (p < 0.05) for at least one target configuration. Bigger and darker dots correspond to ROIs that are consistently significant across multiple target sets. Cool (warm) colors stand for negative (positive) correlations, white identifies ROIs for which the trend changes for different target sets. The ROI with a thick black border is the left middle temporal temporooccipital (LMTGtp) region. This region is located in the temporal system, and it is known to be involved in the default mode network. It is the region presenting the strongest negative correlation among the ones consistent for all six target systems. Panel b) shows the scatter plot of τ and age for LMTGtp when the target is the occipital system. [START_REF] Zhang | Distinct individual differences in default mode network connectivity relate to offtask thought and text memory during reading[END_REF]. The other regions are in the temporal system (LMTGad, LPT1, LMTGtp, RMTGpd, RITGpd2), and three of them (LMTGad, LMTGtp, RMTGpd) belong to the default mode network (DMN) [START_REF] Davey | Exploring the role of the posterior middle temporal gyrus in semantic cognition: Integration of anterior temporal lobe with executive processes[END_REF][START_REF] Xu | Activation and Connectivity within the Default Mode Network Contribute Independently to Future-Oriented Thought[END_REF]. Among those, the left middle temporal temporooccipital (LMTGtp) region has the strongest negative correlation, the corresponding scatter plot of the τ and age is reported in Figure 5.9b.

These results suggest that most brain regions, and in particular those located in the temporal system, tend to decline with age. We found that, in general, systems in the same hemisphere as the lesion suffer a larger loss in step-wise target control centrality, 5.10m-x. In particular, when attacking any system but the sensorimotor, there were considerable losses in the centrality values, 5.10a-c,e-i,k-l. In those configurations, the sensorimotor system itself, which started from high τ values in the original case, underwent the most severe losses. Instead, when attacking the sensorimotor, other systems did not sustain considerable losses in the centrality values, 5.10d,j. Globally the sensorimotor system was the most affected by the stroke simulation and the limbic system was the least affected. When simulating stroke, the hemisphere most impacted is the one with the lesion.

Stroke mostly impacts step-wise target control centrality in

The sensorimotor system appears to be the most affected, while the limbic system is the least affected. Attacking the sensorimotor system led to minor global losses, while attacking any of the other systems led to considerable global losses. Panels a-l) show how the absolute difference in centrality values is distributed on the connectomes at the varying of the target set, when the lesion is on the right (panels a-f), or on the left (panels g-l). Panels m-x) show averaged values of the absolute difference for the left (solid orange circles) and right (dotted blue crosses) hemispheres for the corresponding target set configurations, when the lesion is on the right (panels m-r), or on the left (panels s-x). Error bars stand for the standard error across subjects.

Chapter 5.

Step-wise target control centrality: biological applications 57

Discussion

Distribution of step-wise target control centrality in brain networks with different target sets

Controllability refers to the ability of a dynamical system to move from any initial state to any final state in finite time, when applying suitable inputs to well chosen nodes, the drivers [START_REF] Kalman | Mathematical Description of Linear Dynamical Systems[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. However, in some cases it is neither feasible, nor required to steer the state of the whole system, being more appropriate to exert control on only a subsystem, a subset of the nodes in the network called target set [START_REF] Commault | Functional target controllability of networks: Structural properties and efficient algorithms[END_REF][START_REF] Czeizler | Structural Target Controllability of Linear Networks[END_REF][START_REF] Gao | Target control of complex networks[END_REF].

This may happen, for instance, when developing drug cocktails for multiple target purposes [START_REF] Wu | Network Output Controllability-Based Method for Drug Target Identification[END_REF][START_REF] Yang | Finding multiple target optimal intervention in disease-related molecular network[END_REF].

It is also known that, despite the minimum number of inputs required to achieve control over a system is fixed, the actual composition of the driving set may vary [START_REF] Commault | A Classification of Nodes for Structural Controllability[END_REF][START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF]. Even in the case of simple graphs, there may be a huge number of a priori equivalent configurations [START_REF] Heuberger | The number of maximum matchings in a tree[END_REF][START_REF] Liu | Number of maximum matchings of bipartite graphs with positive surplus[END_REF][START_REF] Wagner | On the number of matchings of a tree[END_REF][START_REF] Zdeborová | The number of matchings in random graphs[END_REF]. Moreover, often in the case of biological networks it is possible to only act on a single node at a time, because of physiological or technical constraints. Thus, the problem of understanding the role played by a single node in driving a part of a system is a relevant question.

In the case of the brain, we may be interested in some subsystems pertaining to specific functions. In this paper, we tested the effect of controlling six different structural systems (frontal, limbic, temporal, sensorimotor, parietal, occipital), denoted as target sets, on the step-wise target control centrality (τ ) [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF]. This centrality measure is based on the Kalman rank condition for controllability, and counts the number of target nodes that can be simultaneously controlled by a single driver.

We found the sensorimotor system to be deeply implicated, having the highest values of τ , independently on the target set. Since the sensorimotor is a primary system, involved in the default mode network (DMN), we could speculate saying that primary systems are well-suited to implement control of specific tasks. For instance, in the case of vision, the primary visual cortex can activate extended visual pathways and prompt visual perception and scene understanding [START_REF] Muldoon | Stimulation-Based Control of Dynamic Brain Networks[END_REF].

Decline of step-wise target control centrality when aging

It is well-known that age is associated with cognitive decline [START_REF] Sala-Llonch | Reorganization of brain networks in aging: A review of functional connectivity studies[END_REF]. There is a consensus among the scientific community in identifying a link between aging and a hypoactivation of the prefrontal network, functional disruptions in precuneus, retrosplenial and posterior cingulate cortices, and a compensatory mechanism involving cortical recruitment [START_REF] Tomasi | Aging and functional brain networks[END_REF].

Age is also among the primary risk factors for neurodegenerative disorders [143, Chapter 5.

Step-wise target control centrality: biological applications 58 185]. For instance, it was shown that Alzheimer's disease patients present a deposition of amyloid plaques in precuneus, retrosplenial and posterior cingulate cortices, which were also found in healthy older adults [START_REF] Oh | β-Amyloid affects frontal and posterior brain networks in normal aging[END_REF][START_REF] Rentz | Cognition, reserve, and amyloid deposition in normal aging[END_REF].

The study of the brain as a network and the analysis of topological centrality measures has recently brought insight to the characterization of brain diseases [START_REF] Stam | Modern network science of neurological disorders[END_REF]. However, in some cases results have been inconsistent and modality-dependent [START_REF] Daianu | Rich club analysis in the Alzheimer's disease connectome reveals a relatively undisturbed structural core network[END_REF][START_REF] Guillon | Disrupted core-periphery structure of multimodal brain networks in Alzheimer's disease[END_REF][START_REF] Yan | Rich club disturbances of the human connectome from subjective cognitive decline to Alzheimer's disease[END_REF]. Thus the question is still open to find a detailed and specific characterization of the aging process.

In this study, we combined information from both functional (fMRI) and structural (DTI) neuroimaging data, and we analyzed the effect of aging on a measure of centrality based on network controllability. We found regions in the temporal system, and in particular some regions in the DMN (LMTGad, LMTGtp, RMT-Gpd), and one in the dorsal attention network (RLOCid3), to be deeply affected by aging, in accordance with previous literature studying both healthy subjects [START_REF] Tomasi | Aging and functional brain networks[END_REF] and Alzheimer's disease patients [START_REF] Guillon | Disrupted core-periphery structure of multimodal brain networks in Alzheimer's disease[END_REF].

In particular, the sensitivity of DMN to aging effects was studied as disruption of cortical networks caused by amyloid deposition [START_REF] Oh | β-Amyloid affects frontal and posterior brain networks in normal aging[END_REF], potential gliosis [START_REF] Reyngoudt | Age-related differences in metabolites in the posterior cingulate cortex and hippocampus of normal ageing brain: A 1H-MRS study[END_REF], reduced resting-state activity [START_REF] Damoiseaux | Reduced resting-state brain activity in the "default network" in normal aging[END_REF], and impaired performance in working memory [START_REF] Sambataro | Age-related alterations in default mode network: Impact on working memory performance[END_REF]. The age-related decline in dopamine neurotransmission may contribute to deterioration of functioning of the dorsal attention network, which makes possible sustained attention, one of the most affected cognitive functions that occurs with aging [START_REF] Filley | Attention and vigilance functions in normal aging[END_REF].

The relation between brain controllability and aging is currently under-explored, yet it may bring new insight to the question. Since previous studies have addressed a different behavior for long-range and short-range functional hubs [START_REF] Tomasi | Aging and functional brain networks[END_REF], it may be of interest to study the effect that target sets at a long or short distance from the driver node have on its controllability properties, taking into account the fact that in order to be controlled simultaneously the targets must be on paths at different lengths from the driver [START_REF] Liu | Control Centrality and Hierarchical Structure in Complex Networks[END_REF].

Effect of lesions on the step-wise target control centrality

In the event of a disease as stroke, or of a lesion in younger ages, it is known that the brain is forced to react to the disruption with a functional reorganization [START_REF] Cheng | Reorganization of functional brain networks during the recovery of stroke: A functional MRI study[END_REF][START_REF] Zhu | Disrupted brain connectivity networks in acute ischemic stroke patients[END_REF].

We studied the effect of a localized attack to one hemisphere of the brain on the step-wise target control centrality. We found that the hemisphere in which occurs the lesion underwent larger losses in the centrality. This may be explained by the fact that the networks we are considering have much more connections within hemispheres than between hemispheres, reflecting the modular structure of the brain [START_REF] Fornito | Fundamentals of Brain Network Analysis[END_REF], see Figure SB4. Thus, removing nodes in the lesion when simulating stroke Chapter 5.

Step-wise target control centrality: biological applications 59 may lead to segregation of the two hemispheres and to major losses to the attacked side.

We also confirmed that the least amount of losses in centrality values are found when damaging the sensorimotor system. This is consistent with the fact that this is a primary system, densely connected [START_REF] Narayanan | Redundancy and Synergy of Neuronal Ensembles in Motor Cortex[END_REF][START_REF] Reich | Independent and Redundant Information in Nearby Cortical Neurons[END_REF][START_REF] So | Redundant information encoding in primary motor cortex during natural and prosthetic motor control[END_REF], and presenting a high structural redundancy [START_REF] Medaglia | Brain and cognitive reserve: Translation via network control theory[END_REF]. In future developments, it may be of interest to study how do the losses associated to a lesion vary with the age of the subjects, to test whether our method is consistent with models of brain connectivity.

Methodological considerations

In this work, we exploited the step-wise target controllability framework [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF], which relies on a linear time invariant (LTI) dynamics and is based on the Kalman rank condition. While it is known that the brain presents a nonlinear dynamics, the study of linear models has proved to be beneficial in improving our understanding [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Liu | Control Principles of Complex Networks[END_REF][START_REF] Tang | Colloquium: Control of dynamics in brain networks[END_REF]. Specifically, the controllability of a linearized model can inform on the controllability of the nonlinear model [START_REF] Slotine | Applied Nonlinear Control[END_REF].

In order to study the controllability of brain networks efficiently, we gave a directionality to the connectomes. Despite the fact that it is not currently possible for neuroimaging techniques to discern the directionality of bundles of axons, it is known that each neuron propagates signals through a well-defined direction from the soma to the end of the axon. Previous efforts to direct a connectome relied on the hypothesis that, given a set of brain networks, edges present for all subjects are the oldest, and any new edge would be directed from the new node to the existing cluster [START_REF] Kerepesi | How to Direct the Edges of the Connectomes: Dynamics of the Consensus Connectomes and the Development of the Connections in the Human Brain[END_REF]. However, a limitation of this method is that it is strictly dependent on the initial set of networks, and the procedure would not be easily scalable. Other methods based on local navigation and communicability were devised to infer the directionality of neural signaling, but since they operate best for nodes connected by longer paths, they are not well-suited to perform inference for structurally connected nodes [START_REF] Seguin | Inferring neural signalling directionality from undirected structural connectomes[END_REF].

Our work was instead inspired by diffusion processes [START_REF] Abdelnour | Functional brain connectivity is predictable from anatomic network's Laplacian eigenstructure[END_REF][START_REF] Avena-Koenigsberger | Communication dynamics in complex brain networks[END_REF][START_REF] Goñi | Exploring the Morphospace of Communication Efficiency in Complex Networks[END_REF][START_REF] Raj | A Network Diffusion Model of Disease Progression in Dementia[END_REF][START_REF] Worrell | Optimized connectome architecture for sensory-motor integration[END_REF], and despite the fact that they do not allow for a parallel distribution of the information, our method preserves hierarchical and modular properties of the connectomes,

Conclusions

In this work, we studied a method to quantify the ability of candidate driver nodes to drive the state of a target set in directed brain networks. Further, we studied how Chapter 5.

Step-wise target control centrality: biological applications 60 this centrality measure is distributed, and how does it change with age and in the event of a lesion. The obtained results reveal in a principled way the systems that are well-suited to drive specific target sets, and the tendency of most brain regions to lose importance as growing old. We hope that further developments of this method and other network controllability measures could contribute to the identification of the key nodes in biological networks to better identify targets of brain stimulation to counteract human diseases.

Methods

Structural and functional brain networks

We used data from the NKI-Rockland database [START_REF] Nooner | The NKI-Rockland Sample: A Model for Accelerating the Pace of Discovery Science in Psychiatry[END_REF], and selected 171 subjects, aged from 5 to 85 [START_REF] Brown | Connected brains and minds-The UMCD repository for brain connectivity matrices[END_REF]. For each subject we had access to both structural (DTI) and functional (fMRI) data. We performed both data acquisition and preprocessing, and the construction of the brain networks as in Guillon et al. [START_REF] Guillon | Disrupted core-periphery structure of multimodal brain networks in Alzheimer's disease[END_REF].

We directed the structural connectomes in order to study their controllability properties in a more efficient way.

Directing structural brain networks

Starting from the DTI network, first we applied a logarithmic transformation (log (x + 1)) to make the weights more homogeneous. Then, we performed a biased random walk [START_REF] Gómez-Gardeñes | Entropy rate of diffusion processes on complex networks[END_REF] to direct each edge from the node with lower strength to the node with higher strength. The probability P ji to go from node i to node j, can be computed as

P ji = w ji h w ih (5.6)
where [w] ji is the symmetric, weighted adjacency matrix of the structural network. For each pair of nodes i and j, we had two directed edges with weights P ij and P ji . An example of the resulting matrices is shown in Figure SB5. We chose one direction by keeping only the highest probability (we kept both if they were equal).

At last, we applied a topological criterion for filtering information [START_REF] Fallani | A Topological Criterion for Filtering Information in Complex Brain Networks[END_REF], to lower the connection density of the networks. We considered for each subject networks with mean degree k varying from 1 to 14.

Stroke simulation

We simulated stroke by producing a localized lesion in one hemisphere. We attacked brain regions either in the left or right hemisphere of a system, removing the corresponding nodes in the network. To study differences in centrality values between hemispheres, we considered the same six target systems, but damaging one side. For instance, for the frontal target system we computed the centrality values in two cases: when attacking the right hemisphere, obtaining a lesion in the right frontal system and using as target the left frontal system, and when attacking the left hemisphere. This procedure had not only the effect to lower the number of nodes in the target set, but also to change the topology of the network, disrupting connections to and from the damaged areas.

Step-wise target controllability

We implemented the step-wise target controllability framework [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF], that analyses the single-input target controllability problem, in which the interest is to study the role of a single driver node in controlling a target set of the system.

We assume the linear time invariant (LTI) dynamics

ẋ(n) = Ax(n) + Bu(n), y(n) = Cx(n) (5.7)
where x(n) ∈ R N describes the state of each node at time n, A is the adjacency matrix of the network, B ∈ R N specifies the driver node that will receive an external input, u(n) ∈ R N is its external input (or control) signal, y(n) ∈ R S is the output vector, and C ∈ R S×N is the output matrix identifying the target nodes.

We say that such a system is controllable if it can be guided from any initial state to any desired final state in finite time, with a suitable choice of input. Consistently, the driver can control the target nodes identified in matrix C if it is possible to guide the state of the target nodes from any initial state to any desired final state in finite time, with a suitable choice of the input injected in the driver node.

In this specific application, the state of the network was represented by the fMRI values, the adjacency matrix corresponded to the directed structural connectome for each subject (averaged across mean degree k from 1 to 14), the target sets were Chapter 5.

Step-wise target control centrality: biological applications 62 the structural systems (frontal, limbic, temporal, sensorimotor, parietal, occipital, Figure SB1), ROIs in the target system were ranked according to the mean strength across subjects in the functional connectome.

The step-wise target control centrality (τ ) computes how many target nodes can be controlled simultaneously by a single driver, and provides the highest-ranked controllable configuration of targets.

Self-regulation score To evaluate the contribution of each structural system to the step-wise target control centrality, and in particular to account for how much nodes in the target system could influence other target ROIs, we devised a selfregulation score. Self-regulation for a system was computed as the ratio between average τ in that system and total average τ . In particular, the self-regulation score for the target system is

1 |T | i∈T (τ i ) 1 N j∈V (τ j ) (5.8)
where T is the target set, |T | its cardinality, that is the number of elements in the set, V is the set of all N ROIs in the brain network, and τ i is the step-wise target control centrality of each node i.

Self-regulation above (below) 1 indicates that nodes in the considered system can control more (less) targets than an average node.

Acknowledgments

The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6), and from the Inria Project Lab Program (project Neuromarkers). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.

Chapter 6

Conclusions

In this thesis, we studied the problem of controlling complex networks from a practical perspective, combining both theoretical and experimental aspects. We proposed an optimized centrality measure based on network controllability to have a more robust estimate of the driver nodes and their "controlling" power. This led us to uncover relevant functional interactions between nodes in two different biological applications. In the case of macrophage activation in multiple sclerosis, our results indicate the presence of causal relations between driver nodes and their controllable targets, suggesting our method as a possible alternative to traditional dimensionality reduction techniques. In the case of aging brain networks, we were able to identify brain regions previously implicated in degenerative processes.

Taken together these results suggest the potential in the application of solutions inspired by controllability theory to biological systems. However, careful attention should be paid to the technical details of the implementation, and more research is needed to generalize these principles.

Recent advances in both network theory and controllability theory inspired questions concerning the fundamental controllability principles of complex networks, and their balance between structure, control and function [START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][START_REF] Barzel | Universality in network dynamics[END_REF][START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF][START_REF] Tang | Colloquium: Control of dynamics in brain networks[END_REF]. These kinds of questions have driven the research for a better understanding of biological systems such as the brain [START_REF] Muldoon | Stimulation-Based Control of Dynamic Brain Networks[END_REF][START_REF] Yan | Network control principles predict neuron function in the Caenorhabditis elegans connectome[END_REF], gene regulatory networks [START_REF] Ravindran | Identification of critical regulatory genes in cancer signaling network using controllability analysis[END_REF], and protein-protein interaction networks [START_REF] Ravindran | Network controllability analysis of intracellular signalling reveals viruses are actively controlling molecular systems[END_REF][START_REF] Vinayagam | Controllability analysis of the directed human protein interaction network identifies disease genes and drug targets[END_REF]. Our work can be seen as a step forward in the direction of adopting a control-theoretical perspective to biological applications.

We proposed the step-wise target control centrality to study the role of single driver nodes in influencing specific target sets in a network. We addressed the singleinput target control problem, providing an exact method able to efficiently identify a controllable configuration of targets in sparse, directed networks.

Our method relies on the strong hypothesis that the system can be described

A.2 Supplementary tables

5 . 1

 51 2047. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 iv LIST OF FIGURES v Working principle of step-wise target controllability. Panel a) il-

32 5 . 2

 52 Molecular network and gene activation associated with the proinflammatory state of macrophages. Panel a) shows the molecular network reconstructed through ontology-based techniques from the macrophage.

33 LIST OF FIGURES vi 5 . 3

 3353 lines represent group-averaged values, while transparent patches stand for standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gene network step-wise target control centrality and analysis of robustness for the driver nodes. In panel a) the size of the nodes codes the step-wise target control centrality (τ ) values. Nodes with τ = 0 are classified as not-drivers and are represented in gray. The inset shows that τ values cannot be merely predicted by node degree k (Spearman rho 0.18, p < 0.07). Panel b) shows the percentage of driver nodes (τ > 0) that are lost when removing nodes in a random fashion (black circles), or preferentially attacking high-degree (blue diamonds) or low-degree nodes (light blue triangles). Panel c) shows the percentage of driver nodes that are lost when randomly rewiring (black circles), adding (blue diamonds) or removing edges (light blue triangles). . . . . . . . . . . . . . . . . . . . 35 5.4 Altered driver-target coactivation in multiple sclerosis. Panel a)

38 LIST OF FIGURES vii 5 . 6

 3856 step-wise target control centrality (τ ) values. For target genes, circle sizes represent the number of driver nodes that can control them. Panel b) Venn diagram showing a decrease in number of driver-target coactivations in the multiple sclerosis (MS) patients as compared to HC. In both groups, these functional interactions tend to predominantly involve signaling genes. Panel c) subnetwork of the walks from all the drivers coactivated with the target IFNA1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 Pooled visualization of dysregulated genes along differentially coactivated driver-target walks. Highlighted genes indicate all nodes on walks between coactivated driver-target pairs, either in the healthy control group, or in the multiple sclerosis patients group. Dysregulated genes are shown in red. Edge thickness is proportional to the number of times they are traversed by walks connecting a driver to a target node (information not reported here). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dyregulated drivers and coactivation switch for SOCS-genes. The subnetwork includes all dysregulated drivers (IRF8, NFKB1, SOCS1, SOCS3, TLR7) and their controllable targets. Panel a) shows coactivated pairs for healthy controls (HC), panel b) shows coactivated pairs for multiple sclerosis (MS) patients. A coactivation switch can be appreciated between the healthy control (HC) and MS groups. SOCS1 and SOCS3 are respectively coactive and silent in HC, while they invert their role in the MS group.

52 LIST OF FIGURES viii 5 . 9

 5259 Panels a-f) show how self-regulation scores are distributed on the connectomes at the varying of the target set. Panels g-l) show numerical values of the self-regulation score for the corresponding target sets. Error bars stand for standard error across subjects. The light purple marker denotes the target system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial correlation between step-wise target control centrality (τ )

(

  M0) macrophage activation state. . . . . . . . . . . . . . . . . . . . . . SA9 Log-transformed gene expression for healthy controls (HC), in the 'proinfalammatory' (M1) macrophage activation state. . . . . . . . . . . . . . SA10Log-transformed gene expression for multiple sclerosis (MS) patients, in the 'alert' (M0) macrophage activation state. . . . . . . . . . . . . . . . . . SA11Log-transformed gene expression for multiple sclerosis (MS) patients, in the 'pro-inflammatory' (M1) macrophage activation state. . . . . . . . . . . . x LIST OF TABLES xi SA12Values of ∆ for each gene (5.1.6). The value of the threshold at the 75th percentile is 0.438. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB1 Denomination of regions of interest (ROIs) in the left hemisphere. . . . . . SB2 Denomination of regions of interest (ROIs) in the right hemisphere. . . . . SB3 F-values for the one-way anova test in the case of step-wise target control centrality and self-regulation. . . . . . . . . . . . . . . . . . . . . . . . . SB4 Spearman partial correlation values computed for step-wise target control centrality and age, corrected for outdegree, for regions of interest (ROIs) in the left hemisphere. Missing values correspond to cases in which the step-wise target control centrality is zero for all subjects. . . . . . . . . . SB5 Spearman partial correlation p-values computed for step-wise target control centrality and age, corrected for outdegree, for regions of interest (ROIs) in the left hemisphere. Missing values correspond to cases in which the step-wise target control centrality is zero for all subjects. . . . . . . . . . SB6 Spearman partial correlation values computed for step-wise target control centrality and age, corrected for outdegree, for regions of interest (ROIs) in the right hemisphere. Missing values correspond to cases in which the step-wise target control centrality is zero for all subjects. . . . . . . . . . SB7 Spearman partial correlation p-values computed for step-wise target control centrality and age, corrected for outdegree, for regions of interest (ROIs) in the right hemisphere. Missing values correspond to cases in which the step-wise target control centrality is zero for all subjects. . . . . . . . . .

1 Chapter

 1

( a )

 a Initial and target state. (b) External input on a node, [98].

Figure 3 . 1 :

 31 Figure 3.1: The intuition behind the controllability property of a system refers to the fact that (a) any initial state can be steered to any final state by choosing the input appropriately and (b) applying an external input to a node (here X 1 ), it is possible to constrain the state of the whole network.

Figure 4 . 1 :

 41 Figure 4.1: Methodological validation of step-wise target controllability. We

Chapter 5 . 32 Figure 5 . 1 :

 53251 Figure 5.1: Working principle of step-wise target controllability. Panel a) illus-trates a network with one driver and a target set T = {t 1 , t 2 , t 3 } of cardinality S = 3. The Kalman condition informs us that only two targets are controllable from the driver, i.e. τ = rank(Q T ) = 2. However, there might be up to 3 equivalent configurations that are controllable, i.e. {t 1 , t 2 }, {t 1 , t 3 }, and {t 2 , t 3 }. For larger networks, the number of Kalman tests to perform can be prohibitive, i.e. ( S τ ). Panel b). By introducing a hierarchy among the target nodes, our step-wise method identifies the configuration with the most important nodes by performing only S tests (see Methods). In this example, the first step considers the subgraph containing all the walks from the driver to the target set T = {t 1 }. The associated controllability matrix has full rank, i.e. rank(Q T ) = 1. The first target is therefore retained and the algorithm moves to Step 2, by constructing a new subgraph containing the walks from the driver to the target set T = {t 1 , t 2 }. The rank of the new controllability matrix is now deficient and t 2 is not retained. In Step 3, the new subgraph contains the walks from the driver to T = {t 1 , t 3 }. Because rank(Q T ) is full and there are no more targets, the algorithm stops and returns the controllable configuration t 1 , t 3 .

Chapter 5 . 33 Figure 5 . 2 :

 53352 Figure 5.2: Molecular network and gene activation associated with the proinflammatory state of macrophages. Panel a) shows the molecular network reconstructed through ontology-based techniques from the macrophage.com repository[START_REF] Raza | A logic-based diagram of signalling pathways central to macrophage activation[END_REF][START_REF] Robert | Macrophages.com: An on-line community resource for innate immunity research[END_REF]. The network consists of N = 101 nodes corresponding to genes involved in inflammation; for the sake of interpretation, they are organized in four classes, depending on their function in the cell. Sensing genes are in the membrane of the cell and start a signaling pathway inside the cell, to the transcription factors, which promote the production of secreted molecules. There are L = 211 directed edges representing either activation or inhibition interactions between molecules (Methods). The size of the nodes is proportional to their total degree k. Panel b) shows gene activation computed as the ratio in expression between the "pro-inflammatory" and "alert" states, based on our RNA sequencing data, generated from monocyte-derived macrophages from blood samples of multiple sclerosis patients (n = 8) and healthy controls (n = 8) (Methods). Solid lines represent group-averaged values, while transparent patches stand for standard deviation.

Chapter 5 .

 5 Step-wise target control centrality: biological applications 34 target set (Figure5.3a).

Chapter 5 . 35 Figure 5 . 3 :

 53553 Figure 5.3: Gene network step-wise target control centrality and analysis ofrobustness for the driver nodes. In panel a) the size of the nodes codes the step-wise target control centrality (τ ) values. Nodes with τ = 0 are classified as not-drivers and are represented in gray. The inset shows that τ values cannot be merely predicted by node degree k (Spearman rho 0.18, p < 0.07). Panel b) shows the percentage of driver nodes (τ > 0) that are lost when removing nodes in a random fashion (black circles), or preferentially attacking high-degree (blue diamonds) or low-degree nodes (light blue triangles). Panel c) shows the percentage of driver nodes that are lost when randomly rewiring (black circles), adding (blue diamonds) or removing edges (light blue triangles).

Chapter 5 . 36 Figure 5 . 4 : 37 (Figure 5 .

 53654375 Figure 5.4: Altered driver-target coactivation in multiple sclerosis. Panel a)reports the coactive driver-target pairs, computed as significant Spearman correlations (p< 0.05) between the gene activation of controllable driver-target pairs, for the healthy control (HC) (blue squares) and the multiple sclerosis (MS) (red squares) groups. White squares indicate that there is a controllable walk from the driver to the target, but that their correlation is not significant. Grey squares mean that there is no controllable walk for driver-target pairs. The size of the circles for driver nodes codes for their step-wise target control centrality (τ ) values. For target genes, circle sizes represent the number of driver nodes that can control them. Panel b) Venn diagram showing a decrease in number of driver-target coactivations in the MS patients as compared to HC. In both groups, these functional interactions tend to predominantly involve signaling genes. Panel c) subnetwork of the walks from all the drivers coactivated with the target IFNA1.

Figure 5 . 6 . 38 Figure 5 . 5 :

 563855 Figure 5.6. By looking at how driver and target nodes are differently coactivated in healthy controls and MS patients, we obtain a much clearer description of the gene dysregulation effects. First, many of the previous results can be now appreciated in finer detail, such as i) the reduction in number of coactivated driver-target pairs in MS, ii) the large number of targets that can be controlled by SOCS1 and SOCS3, and iii) the potential of NFKB1, IFNA1 and IFNB1 to affect the driver-target functional interactions.Second, we report an interesting mechanism involving the drivers with the high-

Chapter 5 . 39 Figure 5 . 6 :

 53956 Figure 5.6: Dyregulated drivers and coactivation switch for SOCS-genes. The subnetwork includes all dysregulated drivers (IRF8, NFKB1, SOCS1, SOCS3, TLR7) and their controllable targets. Panel a) shows coactivated pairs for healthy controls (HC), panel b) shows coactivated pairs for multiple sclerosis (MS) patients. A coactivation switch can be appreciated between the HC and MS groups. SOCS1 and SOCS3 are respectively coactive and silent in HC, while they invert their role in the MS group.

. 5 )

 5 where µ M S and µ HC are group-averages for MS patients and healthy controls, respectively, of the gene activation. Nodes with higher ∆ absolute values are ranked first. Highly positive ∆ values indicate a too strong inflammatory response (overactivation) in the MS patients with respect to the healthy controls. Highly negative ∆ values indicate a too weak inflammatory response (under-activation). We define dysregulated genes along the controllable driver-target walks as those for which |∆| is above the 75th percentile.

  suggestions. The research leading to these results has received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6), and from the Inria Project Lab Program (project Neuromarkers). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies. Chapter 5.

51 Figure 5 . 7 :

 5157 Figure 5.7: Step-wise target control centrality (τ ) averaged across systems and subjects, for the limbic and sensorimotor target sets. Panel a)shows τ centrality values for each brain region when the limbic system is the target set. This case is also exemplary of the case in which one among the frontal, temporal, parietal, or occipital system is the target set. In these configurations, the sensorimotor system has the largest values of centrality, suggesting that it may be well-suited to control the other systems. On the other hand, for all target configurations the limbic system presents low centrality values. Panel b) shows τ centrality values for each brain region when the sensorimotor system is the target set. In this case the values are extremely low for all structural systems, suggesting that the sensorimotor system may be difficult to control.

the distribution of step-wise target control centrality are shown in Figure 5 . 7 .

 57 We found that the sensorimotor system has the largest centrality values for all target sets, indicating the ability of single sensorimotor brain regions to control large parts of the other systems, see for instance Figure 5.7a.

Chapter 5 . 54 Figure 5 . 9 :

 55459 Figure 5.9: Partial correlation between step-wise target control centrality (τ )

the damagedhemisphere

  We then ask to what extent the centrality values change in the event of a brain attack, or stroke, to the network. To answer this question, we considered the same brain network dataset and we simulated lesions by damaging part of the structural Chapter 5. Step-wise target control centrality: biological applications 55 systems localized in each hemisphere In particular, in brain networks we removed nodes and their associated edges corresponding to the lesion. This led us to consider twelve lesions, left and right sides for the six structural systems, see Figure SB1, Methods. We computed for each node the step-wise target control centrality (τ ) values, and averaged them across systems in the left and right hemisphere, and across subjects. Then, for each ROI we computed the absolute difference between step-wise target control centrality (τ ) in the stroke-simulated case, and original τ , shown in Figure 5.10.

Chapter 5 . 56 Figure 5 . 10 :

 556510 Figure 5.10: Absolute difference in step-wise target control centrality (τ ) be-tween the simulation of a lesion in one hemisphere of the brain and the normal case. When simulating stroke, the hemisphere most impacted is the one with the lesion. The sensorimotor system appears to be the most affected, while the limbic system is the least affected. Attacking the sensorimotor system led to minor global losses, while attacking any of the other systems led to considerable global losses. Panels a-l) show how the absolute difference in centrality values is distributed on the connectomes at the varying of the target set, when the lesion is on the right (panels a-f), or on the left (panels g-l). Panels m-x) show averaged values of the absolute difference for the left (solid orange circles) and right (dotted blue crosses) hemispheres for the corresponding target set configurations, when the lesion is on the right (panels m-r), or on the left (panels s-x). Error bars stand for the standard error across subjects.

Chapter 5 .

 5 Indegree and outdegree of the resulting structural systems averaged across mean degree k and across subjects are shown in Figure SB6. Notice that, by construction, indegree is almost perfectly negatively correlated with outdegree (Spearman rho = -0.97, p < 10 -6 ). Step-wise target control centrality: biological applications 61 Density of the brain networks The subject with the lowest number of connections after applying the procedure to direct the connectomes presented 2350 edges, that is a density of E N (N -1) = 2350 158•157 = 0.0947. Since N •k N (N -1) = 15 157 = 0.0955, it was not possible to consider mean degrees above 14. With a mean degree of k = 14, the maximum density considered was N •k N (N -1) = 14 157 = 0.0892, and the minimum density considered was N •k N (N -1) = 1 157 = 0.0064.

Figure SA2 :

 SA2 Figure SA2: Subnetwork illustrating the feedback cycle between dysregulated genes IFNA1, IFNB1 and NFKB1. The three nodes belong to the only strongly connected component (a subnetwork in which every node is reachable from any other node) of the network having more than two nodes. It plays, thus, a central role in the network topology.

Figure SB2 :

 SB2 Figure SB2: Step-wise target control centrality averaged across systems and subjects, for different target sets. Panels a-f) show how the step-wise target control centrality (τ ) is distributed on the connecomes at the varying of the target set. Panels g-l) show numerical values of the centrality for the corresponding target sets. Error bars stand for standard deviation across subjects.

Figure SB3 :

 SB3 Figure SB3: Partial correlation between step-wise target control centrality and age, corrected for outdegree, for different target sets. Only significant (p < 0.05) values are shown.

Figure SB4 :

 SB4 Figure SB4: Number of edges within and between hemispheres.As the mean degree increases, the gap between the number of edges within and between hemispheres becomes more and more accentuated. LL denotes edges within the left hemisphere, RR denotes edges within the right hemisphere, and LR+RL denotes edges between hemispheres.

Figure SB5 :

 SB5 Figure SB5: Structural connectome retrieved from the diffusion tensor imaging (DTI) data and the matrix resulting from the biased random walk process, for an exemplary subject. Panel a) represents the adjacency matrix of the DTI connectome; panel b) the output of the biased random walk process. Panel c) shows a histogram for the node-strength of the original case; panel c) shows a histogram for the node-strength obtained from the biased random walk.

Figure SB6 :

 SB6 Figure SB6: Indegree and outdegree of the structural systems, computed on the directed connectomes and averaged across mean degree k from 1 to 14, and across subjects. Error bars stand for standard deviation across subjects.

  1,2 , Jennifer Fransson 1 , Vincent Henry 1,2 , Olivier Colliot 1,2 , Violetta Zujovic 1 , Fabrizio De Vico Fallani 1,2,*

	1 Sorbonne Université, UPMC Univ Paris 06, Inserm U-1127, CNRS UMR-7225,
	Institut du Cerveau, Hopital Pitié-Salpêtrière, Paris, France
	2 Inria Paris, Aramis Project Team, Paris, France

Step-wise target control centrality: biological applications 45

  Repeat until termination criteria are met. For j ← 1, ..., S do -Add the j-th target node to the target set T ← T ∪ {t j }

	procedure for each candidate driver node
	• Step 1. Initialization
	-Create a temporary empty target set T ← {}
	-Set the number of controllable targets τ ← 0
	• Step 2.

t 1 t 2 ... t S . Then we perform the following step-wise Chapter 5. -Build the subgraph containing the nodes on walks from the driver to the targets in T -Compute the rank of the target controllability matrix Q T

Table SA1 :

 SA1 List of all node genes and associated class depending on their functional role.

	Centrality Driver	Controllable targets
	10	SOCS1	CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IFNG; IL12B; IL1B; NOS2
	10	SOCS3	CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IFNG; IL12B; IL1B; NOS2
	9	IFNGR1	CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2
	9	Gene IFNGR2	Class CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2 Gene (cont.) Class (cont.)
	9	AIFM1 JAK1	transcription CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2 IL15 secreted molecule
	9 9 9 8 8	APAF1 JAK2 ATF2 PRKCD BAK1 STAT1 BAX IRF1 PRKACA CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B signaling IL1B CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2 secreted molecule transcription IRF1 CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2 transcription signaling IRF2 CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B; NOS2 transcription signaling IRF3 CCL5; CXCL10; CXCL11; IFNA1; IFNB1; CXCL9; IL12B; IL1B transcription
	7	BBC3 IRF2	transcription CCL5; CXCL10; CXCL11; IFNA1; IFNB1; IL1B; NOS2 IRF4 transcription
	1	BCL2 CASP1	signaling IL12B	IRF5	transcription
	1	BCL2L1 IRF8	signaling IL12B	IRF7	transcription
	1	BID SPI1	signaling IL12B	IRF8	transcription
		BIRC2	transcription	IRF9	transcription
		BIRC3	transcription	JAK1	signaling
		CASP1	signaling	JAK2	signaling
		CASP10	signaling	MDM2	signaling
		CASP2	signaling	MYD88	signaling
		CASP3	signaling	NFKB1	transcription
		CASP4	signaling	NFKB2	transcription
		CASP6	signaling	NFKBIA	transcription
		CASP7	signaling	NOS2	secreted molecule
		CASP8	signaling	OAS1	signaling
		CASP9	signaling	PARP1	signaling
		CCL5	secreted molecule PLSCR1	transcription
		CFLAR	transcription	PRKACA	signaling
		CHUK	signaling	PRKCD	signaling
		CIITA	transcription	PRKRA	signaling
		CXCL10	secreted molecule PSMB9	secreted molecule
		CXCL11	secreted molecule RELA	transcription
		CXCL9	secreted molecule RELB	transcription
		CYCS	signaling	RIPK1	signaling
		DFFA	signaling	SOCS1	signaling
		DIABLO	signaling	SOCS3	signaling
		EIF2AK2 signaling	SOD2	signaling
		ENDOG	transcription	SP100	secreted molecule
		EP300	transcription	SPI1	transcription
		FADD	signaling	STAT1	transcription
		FAS	sensing	STAT2	transcription
		FASLG	signaling	TBK1	signaling
		GBP1	signaling	TICAM1	signaling
		HLAB	sensing	TICAM2	signaling
		ICAM1	sensing	TLR3	sensing
		IFNA1	secreted molecule TLR7	sensing
		IFNAR1	sensing	TLR9	sensing
		IFNAR2	sensing	TNFRSF10A sensing
		IFNB1	secreted molecule TNFRSF10B sensing
		IFNG	secreted molecule TNFSF10	sensing
		IFNGR1	sensing	TP53	transcription
		IFNGR2	sensing	TRADD	signaling
		IKBKB	transcription	TRAF1	signaling
		IKBKE	transcription	TRAF2	signaling
		IKBKG	transcription	TRAF6	signaling
		IL12B	secreted molecule TYK2	signaling
				XIAP	transcription

Table SA2 :

 SA2 Driver nodes and their controllable targets. The majority of driver nodes

	(omitted from the table) are able to control 5 target nodes (CCL5, CXCL10, CXCL11,
	IFNA1, IFNB1).
	Class	Nodes Drivers Fraction of drivers in the class
	Sensing	13	7	0.54
	Signaling	44	20	0.45
	Transcription	31	18	0.58

Table SA3 :

 SA3 Distribution of driver nodes across different classes of genes.

	rho HC	IL12B CXCL11 CXCL10 CXCL9	IL1B	CCL5	IFNA1 IFNG IFNB1 NOS2
	ATF2	-	0.262	-0.024	-	-	0.762	0.381	-	0.190	-
	CASP1	-	-	-	-	0.667	-	-	-	-	-
	CHUK	-	0.643	0.429	-	-	0.286	-0.190	-	0.000	-
	EIF2AK2	-	0.690	0.738	-	-	0.738	0.452	-	0.286	-
	EP300	-	0.524	0.333	-	-	0.619	0.024	-	-0.143	-
	FADD	-	0.381	0.190	-	-	0.833	0.286	-	0.024	-
	IFNAR1	-	0.452	0.143	-	-	0.714	0.381	-	0.167	-
	IFNAR2	-	0.571	0.333	-	-	0.976	0.429	-	0.143	-
	IFNGR1	-0.214	-0.262	-0.524	-0.429	0.071	0.000	0.048	-	0.190	0.204
	IFNGR2	0.833	0.524	0.238	0.262	0.738	0.905	0.333	-	0.024	0.910
	IKBKB	-	-0.048	-0.119	-	-	0.524	0.262	-	-0.119	-
	IKBKE	-	0.405	0.310	-	-	0.738	0.452	-	0.095	-
	IKBKG	-	0.286	0.024	-	-	0.595	0.190	-	0.214	-
	IRF1	0.810	0.595	0.381	0.500	0.905	0.952	0.595	-	0.405	-
	IRF2	-	0.357	0.119	-	0.738	0.905	0.595	-	0.286	0.838
	IRF3	-	-0.167	0.167	-	-	-0.595	-0.286	-	-0.071	-
	IRF5	-	-0.310	-0.190	-	-	0.095	0.119	-	-0.357	-
	IRF7	-	0.929	0.929	-	-	0.643	0.167	-	0.286	-
	IRF8	-	-	-	-	0.357	-	-	-	-	-
	IRF9	-	0.476	0.595	-	-	0.500	0.357	-	0.190	-
	JAK1	0.476	-0.024	-0.262	-0.143	0.381	0.571	0.167	-	-0.167	0.431
	JAK2	0.595	0.810	0.667	0.548	0.738	0.690	0.333	-	0.310	0.778
	MYD88	-	-0.095	0.143	-	-	-0.429	0.286	-	0.476	-
	NFKB1	-	0.286	0.071	-	-	0.786	0.310	-	0.048	-
	PRKACA	-0.833	-0.476	-0.357	-0.190	-0.548 -0.714	-0.024	-	0.310	-
	PRKCD	0.143	-0.119	-0.405	-0.190	0.286	0.381	0.190	-	0.024	0.431
	PRKRA	-	-0.548	-0.381	-	-	-0.810	-0.571	-	-0.500	-
	RELA	-	0.190	-0.071	-	-	0.786	0.405	-	0.119	-
	RIPK1	-	0.333	0.167	-	-	0.881	0.643	-	0.333	-
	SOCS1	0.690	0.500	0.238	0.381	0.714	0.857	0.452	0.228	0.167	0.898
	SOCS3	0.357	0.119	-0.143	0.238	0.619	0.571	0.214	0.736	0.167	0.323
	SPI1	-	-	-	-	-0.452	-	-	-	-	-
	STAT1	0.214	0.643	0.762	0.762	0.310	0.190	0.095	-	0.190	0.156
	STAT2	-	0.738	0.905	-	-	0.357	0.310	-	0.381	-
	TBK1	-	0.476	0.262	-	-	0.833	0.595	-	0.476	-
	TICAM1	-	0.143	0.238	-	-	0.429	0.143	-	-0.214	-
	TICAM2	-	0.571	0.310	-	-	0.810	0.286	-	0.214	-
	TLR3	-	0.476	0.643	-	-	0.452	0.500	-	0.286	-
	TLR7	-	0.286	0.595	-	-	-0.262	0.048	-	0.214	-
	TLR9	-	-0.310	-0.190	-	-	-0.690	-0.452	-	-0.381	-
	TP53	-	-0.024	0.262	-	-	-0.476	-0.071	-	0.000	-
	TRADD	-	0.190	-0.048	-	-	0.643	0.000	-	-0.310	-
	TRAF2	-	0.667	0.381	-	-	0.833	-0.024	-	-0.190	-
	TRAF6	-	0.214	-0.095	-	-	0.714	0.119	-	-0.167	-
	TYK2	-	-0.143	0.095	-	-	-0.786	-0.595	-	-0.238	-

Table SA4 :

 SA4 Spearman cross-correlation values for healthy controls (HC) for controllable driver-target node pairs. Values for pairs that were not controllable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table.

	p values HC IL12B CXCL11 CXCL10 CXCL9	IL1B	CCL5 IFNA1 IFNG IFNB1 NOS2
	ATF2	-	0.536	0.977	-	-	0.037	0.360	-	0.665	-
	CASP1	-	-	-	-	0.083	-	-	-	-	-
	CHUK	-	0.096	0.299	-	-	0.501	0.665	-	1.000	-
	EIF2AK2	-	0.069	0.046	-	-	0.046	0.267	-	0.501	-
	EP300	-	0.197	0.428	-	-	0.115	0.977	-	0.752	-
	FADD	-	0.360	0.665	-	-	0.015	0.501	-	0.977	-
	IFNAR1	-	0.267	0.752	-	-	0.058	0.360	-	0.703	-
	IFNAR2	-	0.151	0.428	-	-	0.000	0.299	-	0.752	-
	IFNGR1	0.619	0.536	0.197	0.299	0.882	1.000	0.935	-	0.665	0.629
	IFNGR2	0.015	0.197	0.582	0.536	0.046	0.005	0.428	-	0.977	0.003
	IKBKB	-	0.935	0.793	-	-	0.197	0.536	-	0.793	-
	IKBKE	-	0.327	0.462	-	-	0.046	0.267	-	0.840	-
	IKBKG	-	0.501	0.977	-	-	0.132	0.665	-	0.619	-
	IRF1	0.022	0.132	0.360	0.216	0.005	0.001	0.132	-	0.327	-
	IRF2	-	0.389	0.793	-	0.046	0.005	0.132	-	0.501	0.013
	IRF3	-	0.703	0.703	-	-	0.132	0.501	-	0.882	-
	IRF5	-	0.462	0.665	-	-	0.840	0.793	-	0.389	-
	IRF7	-	0.002	0.002	-	-	0.096	0.703	-	0.501	-
	IRF8	-	-	-	-	0.389	-	-	-	-	-
	IRF9	-	0.243	0.132	-	-	0.216	0.389	-	0.665	-
	JAK1	0.243	0.977	0.536	0.752	0.360	0.151	0.703	-	0.703	0.285
	JAK2	0.132	0.022	0.083	0.171	0.046	0.069	0.428	-	0.462	0.030
	MYD88	-	0.840	0.752	-	-	0.299	0.501	-	0.243	-
	NFKB1	-	0.501	0.882	-	-	0.028	0.462	-	0.935	-
	PRKACA	0.015	0.243	0.389	0.665	0.171	0.058	0.977	-	0.462	-
	PRKCD	0.752	0.793	0.327	0.665	0.501	0.360	0.665	-	0.977	0.285
	PRKRA	-	0.171	0.360	-	-	0.022	0.151	-	0.216	-
	RELA	-	0.665	0.882	-	-	0.028	0.327	-	0.793	-
	RIPK1	-	0.428	0.703	-	-	0.007	0.096	-	0.428	-
	SOCS1	0.069	0.216	0.582	0.360	0.058	0.011	0.267	0.583	0.703	0.005
	SOCS3	0.389	0.793	0.752	0.582	0.115	0.151	0.619	0.050	0.703	0.432
	SPI1	-	-	-	-	0.267	-	-	-	-	-
	STAT1	0.619	0.096	0.037	0.037	0.462	0.665	0.840	-	0.665	0.716
	STAT2	-	0.046	0.005	-	-	0.389	0.462	-	0.360	-
	TBK1	-	0.243	0.536	-	-	0.015	0.132	-	0.243	-
	TICAM1	-	0.752	0.582	-	-	0.299	0.752	-	0.619	-
	TICAM2	-	0.151	0.462	-	-	0.022	0.501	-	0.619	-
	TLR3	-	0.243	0.096	-	-	0.267	0.216	-	0.501	-
	TLR7	-	0.501	0.132	-	-	0.536	0.935	-	0.619	-
	TLR9	-	0.462	0.665	-	-	0.069	0.267	-	0.360	-
	TP53	-	0.977	0.536	-	-	0.243	0.882	-	1.000	-
	TRADD	-	0.665	0.935	-	-	0.096	1.000	-	0.462	-
	TRAF2	-	0.083	0.360	-	-	0.015	0.977	-	0.665	-
	TRAF6	-	0.619	0.840	-	-	0.058	0.793	-	0.703	-
	TYK2	-	0.752	0.840	-	-	0.028	0.132	-	0.582	-

Table SA5 :

 SA5 Spearman cross-correlation p-values for healthy controls (HC) for controllable driver-target node pairs. Values for pairs that were not controllable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table.

	rho MS	IL12B CXCL11 CXCL10 CXCL9	IL1B	CCL5	IFNA1	IFNG	IFNB1	NOS2
	ATF2	-	-0.571	-0.833	-	-	-0.143	0.708	-	0.293	-
	CASP1	-	-	-	-	-0.024	-	-	-	-	-
	CHUK	-	0.643	0.476	-	-	0.452	-0.244	-	-0.317	-
	EIF2AK2	-	-0.333	-0.214	-	-	-0.667	0.464	-	0.488	-
	EP300	-	-0.548	-0.119	-	-	-0.452	0.073	-	0.073	-
	FADD	-	0.143	-0.286	-	-	0.619	-0.024	-	0.122	-
	IFNAR1	-	0.262	-0.048	-	-	0.548	0.000	-	-0.268	-
	IFNAR2	-	-0.429	-0.286	-	-	-0.429	0.195	-	-0.024	-
	IFNGR1	0.095	-0.167	0.167	-0.333	0.310	0.095	-0.024	-	0.024	0.190
	IFNGR2	-0.310	-0.071	-0.619	0.095	0.048	0.595	0.049	-	-0.122	0.238
	IKBKB	-	-0.476	-0.381	-	-	-0.643	0.586	-	0.439	-
	IKBKE	-	0.476	-0.048	-	-	0.929	-0.293	-	-0.146	-
	IKBKG	-	0.190	0.048	-	-	0.048	-0.244	-	-0.195	-
	IRF1	0.381	-0.357	-0.595	-0.595	0.524	0.167	0.708	-	0.195	-
	IRF2	-	-0.643	-0.952	-	-0.024 -0.071	0.464	-	0.171	0.452
	IRF3	-	0.452	0.786	-	-	-0.024	-0.683	-	-0.220	-
	IRF5	-	-0.524	-0.452	-	-	-0.500	0.342	-	0.122	-
	IRF7	-	0.500	0.571	-	-	-0.095	-0.171	-	0.293	-
	IRF8	-	-	-	-	0.476	-	-	-	-	-
	IRF9	-	-0.405	-0.143	-	-	-0.762	0.220	-	0.220	-
	JAK1	0.595	-0.238	-0.452	-0.548	0.571	0.048	0.756	-	0.366	0.786
	JAK2	0.190	0.476	0.500	0.357	-0.167	0.238	-0.586	-	0.024	-0.429
	MYD88	-	-0.762	-0.333	-	-	-0.857	0.171	-	-0.244	-
	NFKB1	-	0.071	-0.405	-	-	0.405	0.732	-	0.683	-
	PRKACA	-0.095	0.500	0.810	0.381	0.119	0.214	-0.512	-	-0.171	-
	PRKCD	-0.190	-0.095	-0.381	-0.405	0.595	0.571	0.171	-	-0.073	0.500
	PRKRA	-	0.119	0.571	-	-	-0.381	-0.073	-	0.293	-
	RELA	-	-0.190	-0.714	-	-	0.143	0.683	-	0.464	-
	RIPK1	-	-0.738	-0.714	-	-	-0.524	0.659	-	0.317	-
	SOCS1	0.333	-0.143	-0.190	-0.405	0.143	-0.167	-0.073	-0.275	0.659	0.119
	SOCS3	0.333	-0.048	-0.524	-0.310	0.833	0.429	0.756	0.503	0.366	0.929
	SPI1	-	-	-	-	-0.262	-	-	-	-	-
	STAT1	-0.119	0.571	0.833	0.619	-0.286	0.143	-0.708	-	-0.293	-0.667
	STAT2	-	-0.119	0.190	-	-	-0.714	0.220	-	0.586	-
	TBK1	-	-0.571	-0.714	-	-	-0.429	0.732	-	0.415	-
	TICAM1	-	-0.143	-0.595	-	-	0.119	0.464	-	0.634	-
	TICAM2	-	-0.095	0.048	-	-	-0.286	0.634	-	0.634	-
	TLR3	-	-0.690	-0.286	-	-	-0.286	-0.415	-	-0.488	-
	TLR7	-	-0.286	0.190	-	-	-0.905	0.268	-	0.049	-
	TLR9	-	0.524	0.476	-	-	0.619	-0.586	-	-0.512	-
	TP53	-	0.643	0.714	-	-	0.024	-0.634	-	-0.122	-
	TRADD	-	-0.476	-0.548	-	-	-0.167	0.293	-	-0.122	-
	TRAF2	-	-0.071	-0.619	-	-	0.595	0.049	-	-0.122	-
	TRAF6	-	-0.429	-0.690	-	-	-0.286	0.756	-	0.293	-
	TYK2	-	0.238	0.381	-	-	-0.333	-0.293	-	0.073	-

Table SA6 :

 SA6 Spearman cross-correlation values for multiple sclerosis (MS) patients for controllable driver-target node pairs. Values for pairs that were not controllable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table.

	p values MS IL12B CXCL11 CXCL10 CXCL9	IL1B	CCL5 IFNA1 IFNG IFNB1 NOS2
	ATF2	-	0.151	0.015	-	-	0.752	0.059	-	0.490	-
	CASP1	-	-	-	-	0.977	-	-	-	-	-
	CHUK	-	0.096	0.243	-	-	0.267	0.567	-	0.455	-
	EIF2AK2	-	0.428	0.619	-	-	0.083	0.260	-	0.227	-
	EP300	-	0.171	0.793	-	-	0.267	0.887	-	0.889	-
	FADD	-	0.752	0.501	-	-	0.115	0.979	-	0.795	-
	IFNAR1	-	0.536	0.935	-	-	0.171	1.000	-	0.531	-
	IFNAR2	-	0.299	0.501	-	-	0.299	0.650	-	0.977	-
	IFNGR1	0.840	0.703	0.703	0.428	0.462	0.840	0.979	-	0.977	0.665
	IFNGR2	0.462	0.882	0.115	0.840	0.935	0.132	0.926	-	0.795	0.582
	IKBKB	-	0.243	0.360	-	-	0.096	0.141	-	0.285	-
	IKBKE	-	0.243	0.935	-	-	0.002	0.489	-	0.743	-
	IKBKG	-	0.665	0.935	-	-	0.935	0.567	-	0.651	-
	IRF1	0.360	0.389	0.132	0.132	0.197	0.703	0.059	-	0.651	-
	IRF2	-	0.096	0.001	-	0.977	0.882	0.260	-	0.701	0.267
	IRF3	-	0.267	0.028	-	-	0.977	0.071	-	0.615	-
	IRF5	-	0.197	0.267	-	-	0.216	0.415	-	0.795	-
	IRF7	-	0.216	0.151	-	-	0.840	0.701	-	0.490	-
	IRF8	-	-	-	-	0.243	-	-	-	-	-
	IRF9	-	0.327	0.752	-	-	0.037	0.617	-	0.615	-
	JAK1	0.132	0.582	0.267	0.171	0.151	0.935	0.036	-	0.387	0.028
	JAK2	0.665	0.243	0.216	0.389	0.703	0.582	0.141	-	0.977	0.299
	MYD88	-	0.037	0.428	-	-	0.011	0.701	-	0.567	-
	NFKB1	-	0.882	0.327	-	-	0.327	0.047	-	0.071	-
	PRKACA	0.840	0.216	0.022	0.360	0.793	0.619	0.210	-	0.701	-
	PRKCD	0.665	0.840	0.360	0.327	0.132	0.151	0.701	-	0.889	0.216
	PRKRA	-	0.793	0.151	-	-	0.360	0.887	-	0.490	-
	RELA	-	0.665	0.058	-	-	0.752	0.071	-	0.257	-
	RIPK1	-	0.046	0.058	-	-	0.197	0.088	-	0.455	-
	SOCS1	0.428	0.752	0.665	0.327	0.752	0.703	0.887	0.505	0.089	0.793
	SOCS3	0.428	0.935	0.197	0.462	0.015	0.299	0.036	0.209	0.387	0.002
	SPI1	-	-	-	-	0.536	-	-	-	-	-
	STAT1	0.793	0.151	0.015	0.115	0.501	0.752	0.059	-	0.490	0.083
	STAT2	-	0.793	0.665	-	-	0.058	0.617	-	0.137	-
	TBK1	-	0.151	0.058	-	-	0.299	0.047	-	0.320	-
	TICAM1	-	0.752	0.132	-	-	0.793	0.260	-	0.101	-
	TICAM2	-	0.840	0.935	-	-	0.501	0.102	-	0.101	-
	TLR3	-	0.069	0.501	-	-	0.501	0.320	-	0.227	-
	TLR7	-	0.501	0.665	-	-	0.005	0.532	-	0.929	-
	TLR9	-	0.197	0.243	-	-	0.115	0.141	-	0.207	-
	TP53	-	0.096	0.058	-	-	0.977	0.102	-	0.795	-
	TRADD	-	0.243	0.171	-	-	0.703	0.489	-	0.795	-
	TRAF2	-	0.882	0.115	-	-	0.132	0.926	-	0.795	-
	TRAF6	-	0.299	0.069	-	-	0.501	0.036	-	0.490	-
	TYK2	-	0.582	0.360	-	-	0.428	0.489	-	0.889	-

Table SA7 :

 SA7 Spearman cross-correlation p-values for multiple sclerosis (MS) patients for controllable driver-target node pairs. Values for pairs that were not controllable are not reported (-). Targets IL15, PSMB9, and SP100 could not be controlled by any driver and are omitted from the table.

	M0	HC 1	HC 2	HC 3	HC 4	HC 5	HC 6	HC 7	HC 8
	AIFM1	8.905	8.673	8.806	8.798	8.986	8.733	8.940	8.875
	APAF1	8.127	8.284	8.042	7.999	8.149	8.241	8.035	8.017
	ATF2	8.102	8.206	7.941	8.301	8.158	8.219	8.135	8.115
	BAK1	8.312	7.959	8.313	7.894	8.046	7.817	8.002	8.298
	BAX	8.664	8.785	8.716	8.568	8.537	8.836	8.699	8.670
	BBC3	6.621	6.490	6.713	6.305	6.371	6.809	6.360	6.336
	BCL2	6.067	5.871	5.487	5.451	5.663	5.549	5.577	5.891
	BCL2L1	8.831	8.650	8.594	8.545	8.621	8.411	8.591	8.809
	BID	8.553	8.843	8.349	8.772	8.649	8.716	8.771	8.670
	BIRC2	8.601	8.414	8.567	8.408	8.446	8.346	8.370	8.492
	BIRC3	7.605	6.566	7.761	7.774	7.384	6.970	7.109	7.415
	CASP1	8.963	9.305	9.039	9.250	8.894	9.297	9.512	9.071
	CASP10	6.885	6.626	7.219	7.080	7.045	6.937	7.107	6.923
	CASP2	7.721	8.026	7.882	7.950	7.764	7.747	7.692	7.745
	CASP3	7.469	7.325	7.651	7.355	7.418	7.519	7.347	7.400
	CASP4	8.716	9.072	8.963	9.221	8.823	9.078	9.115	8.800
	CASP6	6.380	6.557	6.563	6.575	6.423	6.623	6.694	6.453
	CASP7	7.431	7.448	7.595	7.442	7.444	7.224	7.430	7.491
	CASP8	6.908	7.052	7.136	7.023	6.992	6.955	7.032	6.954
	CASP9	7.080	6.838	7.015	6.985	7.194	6.777	6.827	7.141
	CCL5	6.554	5.376	9.019	9.032	6.152	6.005	6.213	7.192
	CFLAR	8.333	8.279	8.331	8.373	8.333	8.219	8.188	8.214
	CHUK	8.734	8.704	8.624	8.693	8.780	8.697	8.902	8.813
	CIITA	7.211	8.597	7.213	8.157	7.585	8.604	7.922	7.432
	CXCL10	4.324	8.678	10.792	10.791	4.001	6.920	6.396	7.669
	CXCL11	1.994	5.361	9.017	9.042	1.614	3.848	4.821	6.164
	CXCL9	0.000	4.618	10.422	10.434	2.571	3.114	5.543	6.836
	CYCS	10.773	10.679	10.662	10.485	10.762	10.595	10.825	10.828
	DFFA	6.989	7.077	6.902	7.094	7.172	7.249	7.161	7.093
	DIABLO	9.117	8.956	9.019	9.055	8.962	9.023	9.040	9.088
	EIF2AK2	7.651	8.310	8.359	8.474	7.771	8.051	7.874	7.897
	ENDOG	5.237	5.146	4.920	5.369	4.974	5.345	5.078	5.461
	EP300	9.343	9.239	9.200	9.337	9.293	9.353	9.250	9.397
	FADD	8.344	8.090	8.293	8.238	8.345 8.318 0.000 8.132 8.195 0.000 0.000 0.000
	IFNGR1	9.712	10.156	9.524	9.925	10.004	10.483	10.286	9.743
	IFNGR2	9.468	9.551	9.518	9.807	9.561	9.667	9.749	9.521
	IKBKB	8.256	8.423	8.363	8.351	8.366	8.356	8.297	8.220
	IKBKE	8.601	8.705	8.569	8.738	8.685	8.655	8.548	8.767
	IKBKG	7.948	8.004	7.830	8.026	7.924	7.853	7.903	7.920
	IL12B	1.428	2.114	6.206	6.281	0.000	0.000	2.664	3.869
	IL15	5.543	5.210	5.937	5.985	5.212	5.300	5.626	5.489
	IL1B	6.621	7.531	8.790	8.797	6.276	6.501	6.794	7.316
	IRF1	7.233	6.954	7.983	7.994	7.078	6.902	7.068	7.026
	IRF2	7.456	7.574	7.494	7.638	7.492	7.532	7.381	7.273
	IRF3	7.244	7.514	7.473	7.230	7.372	7.427	7.360	7.223
	IRF4	4.292	5.445	5.993	5.990	4.546	5.538	4.062	4.744
	IRF5	9.809	9.975	9.819	9.736	9.724	9.786	9.859	9.828
	IRF7	5.794	7.123	7.423	7.519	5.922	6.612	5.577	6.247
	IRF8	8.265	8.437	8.307	8.542	8.342	8.514	8.423	8.403
	IRF9	7.767	7.972	7.882	7.944	7.632	7.818	7.645	7.742
	JAK1	10.102	9.993	9.917	10.002	9.985	10.059	10.047	9.920
	JAK2	8.466	8.377	8.722	8.734	8.474	8.285	8.365	8.532
	MDM2	9.512	9.220	9.520	9.546	9.117	9.335	9.549	9.510
	MYD88	10.918	11.245	10.748	11.002	10.962	11.123	11.061	10.322
	NFKB1	9.150	9.448	9.171	9.506	9.014	9.394	9.235	9.186
	NFKB2	9.122	8.870	9.078	9.114	8.808	8.652	8.676	9.074
	NFKBIA	8.894	9.422	9.105	9.490	8.948	9.261	9.161	9.107
	NOS2	0.000	0.000	0.889	0.000	0.000	0.000	0.000	0.000
	OAS1	7.541	8.102	8.822	8.776	7.849	8.002	7.805	7.524
	PARP1	8.682	8.816	8.479	8.882	8.425	8.607	8.559	8.508
	PLSCR1	8.923	9.430	9.304	9.365	9.054	9.388	9.102	9.146
	PRKACA	8.796	9.172	8.751	9.017	8.817	9.084	9.019	8.828
	PRKCD	10.392	10.376	10.214	10.307	10.383	10.164	10.354	10.375
	PRKRA	6.953	6.895	6.947	6.842	7.011	6.869	6.983	6.969
	PSMB9	8.163	8.054	8.557	8.382	7.990	7.770	8.089	8.119
	RELA	8.171	8.155	8.200	8.416	8.031	8.044	8.044	8.103
	RELB	7.822	8.005	7.699	8.034	7.624	7.969	7.660	7.825
	RIPK1	7.931	7.844	7.988	8.074	7.764	7.890	7.805	7.788
	SOCS1	0.584	0.593	1.794	1.073	0.000	0.541	0.555	0.000
	SOCS3	9.126	8.404	8.736	8.423	9.041	8.509	8.539	8.719
	SOD2	10.051	9.952	10.284	10.725	9.817	9.910	10.236	9.475
	SP100	8.473	8.590	8.622	8.760	8.382	8.604	8.464	8.422
	SPI1	10.819	10.926	10.741	10.873	10.847	10.936	10.821	10.863
	STAT1	9.942	9.989	10.660	10.692	10.041	9.866	10.046	10.058
	STAT2	7.201	7.616	7.686	7.779	7.285	7.479	7.399	7.275
	TBK1	8.621	8.559	8.668	8.672	8.660	8.490	8.504	8.542
	TICAM1	8.515	8.606	8.544	8.614	8.246	8.546	8.232	8.536
	TICAM2	8.641	8.833	8.668	8.827	8.655	8.882	8.735	8.692
	TLR3	4.324	5.638	6.217	6.423	5.028	5.424	5.330	4.247
	TLR7	6.668	8.073	7.613	7.631	6.495	8.035	7.235	6.941
	TLR9	9.757	9.562	9.636	9.809	9.882	9.727	9.723	9.756
	TNFRSF10A	7.335	7.034	7.109	6.550	7.312	6.661	6.779	6.684
	TNFRSF10B	7.626	7.808	7.614	7.730	7.424	7.709	7.654	7.859
	TNFSF10	5.363	8.369	9.148	9.136	6.083	7.515	6.084	6.469
	TP53	6.919	7.319	7.034	7.381	7.033	7.326	7.224	7.132
	TRADD	5.908	5.954	6.001	5.785	5.828	6.088	5.766	6.001
	TRAF1	7.253	6.321	6.981	6.823	6.677	6.055	5.763	6.689
	TRAF2	7.233	7.010	7.088	7.216	7.164	6.987	7.073	7.306
	TRAF6	6.487	6.259	6.343	6.236	6.315	6.372	6.222	6.343
	TYK2	9.089	9.265	9.118	9.083	9.004	9.144	8.862	9.106
	XIAP	8.123	7.712	7.867	7.850	8.119	7.848	8.059	8.137

Table SA8 :

 SA8 Log

	M1	HC 1	HC 2	HC 3	HC 4	HC 5	HC 6	HC 7	HC 8
	AIFM1	8.615	7.912	8.496	8.189	8.604	7.933	8.398	8.430
	APAF1	8.134	8.186	8.132	8.154	8.382	8.627	8.507	8.296
	ATF2	7.945	8.737	7.993	8.634	8.339	8.801	8.550	8.343
	BAK1	9.053	9.183	8.969	8.673	8.897	8.774	8.964	9.045
	BAX	8.614	9.429	8.808	8.968	8.681	9.187	9.035	9.057
	BBC3	7.053	8.465	7.422	7.664	7.465	8.407	7.976	7.747
	BCL2	5.342	5.820	4.942	5.617	5.191	5.225	5.716	5.372
	BCL2L1	8.735	8.938	8.645	8.724	8.591	8.630	8.537	8.784
	BID	8.673	9.212	8.880	9.267	9.110	9.040	9.356	9.042
	BIRC2	8.675	9.358	8.774	8.988	8.804	8.963	9.113	8.941
	BIRC3	9.389	11.345	9.550	10.864	10.285	9.823	10.476	10.344
	CASP1	9.839	10.421	9.934	10.222	10.157	10.694	10.556	10.148
	CASP10	8.007	8.085	8.269	8.251	8.540	8.199	8.671	8.394
	CASP2	7.627	7.802	7.537	7.741	7.795	7.634	7.707	7.734
	CASP3	8.121	9.214	8.412	8.368	8.731	8.587	8.537	8.358
	CASP4	9.783	10.589	10.008	10.311	10.156	10.936	10.631	10.153
	CASP6	6.340	6.624	6.463	6.489	6.656	6.265	6.620	6.509
	CASP7	8.195	9.605	8.446	8.952	8.864	9.144	9.152	8.705
	CASP8	7.462	7.797	7.426	7.473	7.651	7.668	7.659	7.542
	CASP9	7.379	7.247	7.445	7.097	7.055	6.957	7.091	7.299
	CCL5	11.156	12.785	11.591	12.279	11.382	12.897	12.534	11.795
	CFLAR	9.491	10.691	9.655	10.195	10.124	10.271	10.425	9.977
	CHUK	9.013	8.961	8.882	8.835	9.103	9.143	9.148	9.046
	CIITA	8.812	9.291	8.626	9.379	9.096	9.390	9.307	9.041
	CXCL10	11.964	14.233	12.714	13.661	13.592	14.033	14.225	13.487
	CXCL11	9.803	13.120	11.015	12.314	12.033	12.457	12.861	12.067
	CXCL9	12.872	13.742	13.088	13.765	13.150	14.170	13.569	13.541
	CYCS	10.436	10.027	10.379	9.963	10.432	9.759	10.156	10.411
	DFFA	6.851	7.016	6.807	7.005	7.064	6.987	7.045	6.954
	DIABLO	9.261	9.159	9.147	9.172	9.177	9.244	9.248	9.191
	EIF2AK2	9.425	10.436	9.757	9.967	10.075	9.942	10.421	9.958
	ENDOG	4.881	4.169	4.730	4.778	4.489	4.231	4.191	4.572
	EP300	9.535	9.582	9.496	9.572	9.668	9.657	9.559	9.595
	FADD	8.506	8.916	8.392	8.626	8.689	8.723	8.794	8.596
	FAS	6.854	9.182	7.539	8.416	7.671	9.190	8.487	8.079
	FASLG	2.355	1.523	1.494	2.774	2.551	0.000	4.869	0.713
	GBP1	12.238	13.082	12.494	12.904	12.610	13.314	13.149	12.747
	HLA-B	11.931	11.876	9.179	11.932	5.275	13.018	12.093	7.755
	ICAM1	11.050	11.863	11.072	11.568	11.105	12.021	11.667	11.268
	IFNA1	1.351	2.205	2.806	0.000	2.443	1.682	3.080	3.216
	IFNAR1	9.393	9.386	9.533	9.270	9.621	9.609	9.521	9.504
	IFNAR2	8.401	8.754	8.528	8.516	8.727	8.841	8.821	8.434
	IFNB1	1.067	1.027	1.278	0.000	0.000	1.454	2.250	1.980
	IFNG	0.000	2.101	0.000	4.186	0.000	3.382	5.499	0.000
	IFNGR1	8.797	9.648	9.125	9.455	9.140	10.166	9.787	9.212
	IFNGR2	9.677	10.215	9.826	10.126	10.102	10.310	10.280	9.984
	IKBKB	8.472	8.945	8.560	8.800	8.597	8.678	8.662	8.575
	IKBKE	9.002	9.390	9.071	9.224	9.316	9.168	9.178	9.284
	IKBKG	8.261	8.398	8.124	8.303	8.171	8.323	8.226	8.126
	IL12B	6.411	10.248	7.494	9.974	7.685	7.385	10.435	9.724
	IL15	8.313	8.676	8.145	8.030	8.591	8.771	8.541	8.388
	IL1B	10.393	12.023	10.875	12.141	10.135	12.586	12.451	10.999
	IRF1	10.331	10.758	10.328	10.631	10.411	10.996	10.744	10.495
	IRF2	7.896	8.919	8.142	8.572	8.476	8.833	8.631	8.281
	IRF3	7.967	7.844	7.980	7.938	7.975	7.794	7.971	7.889
	IRF4	7.256	9.826	7.569	9.531	8.217	8.257	9.155	8.257
	IRF5	10.268	10.698	10.219	10.205	10.177	9.915	10.301	10.358
	IRF7	9.010	10.175	9.350	9.847	9.384	9.860	9.382	9.174
	IRF8	8.762	9.621	9.073	9.590	9.119	9.354	9.538	9.310
	IRF9	8.659	8.952	8.657	8.883	8.699	8.729	8.858	8.773
	JAK1	10.210	10.906	10.243	10.761	10.388	10.717	10.672	10.466
	JAK2	10.263	10.297	10.215	10.313	10.527	10.472	10.364	10.556
	MDM2	9.903	10.854	10.129	10.476	10.203	10.520	10.598	10.697
	MYD88	11.025	11.018	10.986	10.994	11.178	11.167	11.239	10.574
	NFKB1	9.771	11.404	10.065	11.028	10.312	11.012	11.088	10.436
	NFKB2	9.901	11.207	10.122	10.851	10.221	10.861	10.778	10.396
	NFKBIA	10.017	11.633	10.306	11.283	10.496	11.667	11.259	10.792
	NOS2	0.000	2.464	1.003	0.000	2.443	2.857	1.060	1.980
	OAS1	9.868	11.017	10.296	10.511	10.695	10.624	10.914	10.442
	PARP1	8.121	8.272	8.058	8.334	8.288	8.263	8.255	8.239
	PLSCR1	10.112	11.014	10.460	10.638	10.573	11.218	10.960	10.498
	PRKACA	8.435	8.217	8.511	8.516	8.262	8.648	8.472	8.424
	PRKCD	10.463	10.719	10.481	10.675	10.582	10.708	10.642	10.677
	PRKRA	6.691	6.484	6.640	6.571	6.611	6.445	6.494	6.595
	PSMB9	10.336	10.679	10.322	10.274	10.322	10.554	10.533	10.256
	RELA	8.772	9.818	8.988	9.624	9.010	9.640	9.490	9.137
	RELB	8.393	9.704	8.638	9.300	8.680	9.448	9.062	8.882
	RIPK1	8.535	9.263	8.616	8.950	8.641	9.036	8.991	8.666
	SOCS1	3.322	4.369	3.324	4.254	3.560	4.419	3.894	3.594
	SOCS3	10.813	11.001	10.406	10.420	10.269	11.389	10.748	10.363
	SOD2	12.142	12.956	12.446	12.876	12.539	13.222	13.149	12.474
	SP100	9.256	10.098	9.440	9.741	9.625	9.960	9.905	9.636
	SPI1	10.787	10.751	10.758	10.807	10.851	10.871	10.595	10.731
	STAT1	12.610	12.253	12.632	12.308	12.685	12.337	12.440	12.563
	STAT2	9.042	9.442	9.363	9.318	9.541	9.316	9.586	9.377
	TBK1	9.271	9.783	9.357	9.508	9.573	9.975	9.810	9.512
	TICAM1	9.197	9.705	9.174	9.416	9.189	9.139	9.283	9.277
	TICAM2	9.120	9.727	9.211	9.534	9.410	9.937	9.654	9.398
	TLR3	5.566	7.439	6.809	7.187	7.051	6.322	7.213	6.757
	TLR7	7.254	7.567	8.002	7.622	8.464	7.550	7.831	8.418
	TLR9	9.765	9.477	9.718	9.775	9.906	9.587	9.582	9.757
	TNFRSF10A	8.508	8.543	8.136	7.733	8.327	8.273	8.267	7.793
	TNFRSF10B	6.938	8.124	7.545	7.951	7.037	8.246	7.987	7.895
	TNFSF10	10.245	11.932	10.954	11.377	11.513	11.587	11.766	11.234
	TP53	6.696	6.907	6.909	7.074	7.273	6.797	6.961	7.162
	TRADD	6.833	7.745	6.877	7.334	7.084	7.556	7.178	6.970
	TRAF1	8.632	10.303	8.590	9.677	9.031	9.706	10.080	9.274
	TRAF2	8.075	8.320	7.872	8.006	8.222	8.189	8.130	8.085
	TRAF6	6.712	7.576	6.730	7.273	7.068	7.459	7.191	7.022
	TYK2	9.658	9.260	9.571	9.383	9.330	9.264	9.155	9.387
	XIAP	8.197	7.905	8.160	7.719	8.153	8.187	8.198	8.144

-transformed gene expression for healthy controls (HC), in the 'alert' (M0) macrophage activation state.

Table SA9 :

 SA9 Log-transformed gene expression for healthy controls (HC), in the 'proinfalammatory' (M1) macrophage activation state.

	M0	MS 1	MS 2	MS 3	MS 4	MS 5	MS 6	MS 7	MS 8
	AIFM1	8.945	8.760	8.623	8.821	8.813	8.844	8.881	9.014
	APAF1	7.751	8.232	8.147	7.699	7.757	8.110	7.877	7.585
	ATF2	8.325	8.163	8.760	8.455	8.493	8.456	8.795	8.833
	BAK1	8.076	7.922	7.808	7.994	8.199	8.031	7.954	8.397
	BAX	8.264	8.578	8.456	8.741	8.836	8.661	8.923	8.489
	BBC3	6.975	6.251	6.297	6.859	6.444	6.372	6.544	6.682
	BCL2	5.709	5.572	5.529	5.663	5.711	5.827	5.215	5.671
	BCL2L1	8.520	8.278	8.253	8.395	8.433	8.093	8.261	8.684
	BID	8.375	8.037	7.533	8.391	7.813	8.014	7.886	8.547
	BIRC2	8.273	8.457	8.396	8.038	8.232	8.136	8.334	8.500
	BIRC3	7.740	6.507	5.240	6.495	5.266	6.242	5.788	7.364
	CASP1	8.584	9.309	9.256	9.195	9.028	9.001	9.274	8.878
	CASP10	6.542	7.054	6.682	6.664	6.880	6.702	7.064	6.403
	CASP2	7.656	8.072	8.342	7.936	7.885	7.991	7.716	7.631
	CASP3	7.192	7.635	7.039	6.799	6.830	6.838	7.026	6.969
	CASP4	9.047	8.940	9.313	9.446	9.269	9.337	9.410	9.252
	CASP6	6.573	6.744	7.086	6.679	7.175	7.068	6.922	7.205
	CASP7	7.309	7.302	7.192	7.314	7.279	6.935	7.098	7.409
	CASP8	6.869	7.229	7.013	6.782	7.057	7.008	7.221	6.858
	CASP9	7.080	7.150	6.372	6.959	6.726	6.530	6.913	7.443
	CCL5	7.329	5.547	5.463	7.867	6.215	7.368	4.360	8.911
	CFLAR	8.343	8.063	7.969	8.075	7.836	7.886	8.075	8.138
	CHUK	8.333	8.562	8.113	8.261	8.141	8.041	8.229	8.472
	CIITA	7.922	7.841	8.436	8.860	8.086	8.954	9.311	7.804
	CXCL10	3.325	2.945	4.194	6.927	3.527	6.803	5.782	4.347
	CXCL11	2.402	0.970	2.401	4.546	0.945	3.486	2.413	1.983
	CXCL9	2.777	2.945	3.974	5.663	3.378	4.887	3.450	5.112
	CYCS	10.139	10.180	9.493	9.987	9.742	9.719	9.833	10.194
	DFFA	7.379	7.008	7.438	7.352	7.598	7.391	7.600	7.482
	DIABLO	9.079	8.957	9.002	8.926	8.874	8.787	8.945	9.160
	EIF2AK2	7.764	7.873	8.174	8.466	8.049	8.581	8.212	7.909
	ENDOG	5.490	4.630	4.766	5.794	5.700	5.656	4.951	5.054
	EP300	9.384	9.157	9.457	9.333	9.324	9.426	9.652	9.493
	FADD	8.154	8.062	7.995	8.034	8.046 8.212 0.000 7.793 7.664 0.000 0.000 0.000
	IFNGR1	9.401	10.078	9.725	9.830	9.657	10.041	9.935	9.584
	IFNGR2	9.738	9.839	10.012	9.810	9.771	9.713	9.507	9.870
	IKBKB	8.484	8.429	8.563	8.536	8.396	8.548	8.551	8.213
	IKBKE	8.668	8.203	7.919	8.767	8.192	8.411	8.237	8.979
	IKBKG	8.249	7.558	7.953	8.283	7.991	8.022	7.934	7.887
	IL12B	0.000	0.970	0.000	0.000	1.422	1.466	1.960	3.374
	IL15	4.942	4.864	5.122	5.128	5.336	5.216	5.263	5.432
	IL1B	7.562	6.378	6.155	7.766	5.962	7.178	6.087	6.870
	IRF1	7.744	7.161	6.963	7.388	7.355	7.078	7.433	7.541
	IRF2	7.696	7.526	7.780	7.623	7.563	7.510	7.697	7.669
	IRF3	7.510	7.576	7.420	7.674	7.551	7.708	7.551	7.343
	IRF4	4.936	4.291	5.122	5.028	3.735	6.220	5.315	5.131
	IRF5	10.234	9.715	9.897	9.960	9.871	9.811	10.118	10.021
	IRF7	6.506	5.491	6.258	7.633	6.495	7.792	6.872	6.445
	IRF8	8.403	8.045	8.089	8.342	8.351	8.185	8.323	8.240
	IRF9	7.860	7.625	7.725	8.023	7.649	7.875	7.915	7.610
	JAK1	10.015	9.965	9.932	9.822	9.784	9.732	9.951	9.759
	JAK2	8.461	8.484	8.034	8.169	8.021	8.393	8.300	8.346
	MDM2	9.056	9.326	8.886	9.221	9.065	8.865	9.767	9.515
	MYD88	11.007	10.799	10.982	11.229	10.798	11.164	11.379	11.402
	NFKB1	8.836	8.829	8.585	9.169	8.780	9.122	8.916	9.008
	NFKB2	9.335	8.660	8.045	9.032	8.373	8.850	7.992	9.103
	NFKBIA	9.368	8.760	8.881	9.626	8.913	9.353	9.247	9.772
	NOS2	0.000	1.453	1.101	1.027	0.945	0.000	0.000	0.000
	OAS1	8.055	7.953	7.740	8.554	7.803	8.187	7.896	7.764
	PARP1	8.157	8.549	8.962	8.753	9.128	8.761	8.593	8.043
	PLSCR1	8.678	8.996	9.030	9.261	8.901	9.291	9.100	9.189
	PRKACA	8.854	8.902	9.006	9.202	8.974	9.104	9.136	8.878
	PRKCD	10.671	10.297	10.200	10.430	10.208	10.267	10.342	10.534
	PRKRA	6.570	6.930	6.854	6.596	6.716	6.611	6.669	6.568
	PSMB9	8.662	7.873	8.275	8.577	8.562	8.899	8.362	8.438
	RELA	8.712	8.052	8.951	8.836	8.780	8.838	9.007	8.943
	RELB	7.768	6.870	6.556	7.707	6.680	7.221	6.607	7.769
	RIPK1	8.063	7.989	8.163	8.040	8.068	7.881	8.140	8.044
	SOCS1	0.608	0.000	0.000	0.640	0.000	1.921	0.701	0.715
	SOCS3	9.059	8.540	7.783	8.766	8.052	8.306	7.712	9.148
	SOD2	10.224	9.102	8.650	9.886	8.712	9.133	9.040	10.282
	SP100	8.427	8.636	8.613	8.549	8.430	8.587	8.606	8.452
	SPI1	11.371	10.573	11.177	11.473	11.286	11.361	11.477	11.426
	STAT1	9.910	9.759	9.290	10.016	9.499	9.729	9.673	9.778
	STAT2	7.512	7.210	7.210	7.753	7.438	7.939	7.653	6.887
	TBK1	8.596	8.508	8.661	8.323	8.526	8.318	8.632	8.508
	TICAM1	8.851	7.904	8.555	8.776	8.327	8.733	8.639	8.891
	TICAM2	8.419	8.702	8.671	8.520	8.589	8.588	8.665	8.355
	TLR3	4.595	4.845	4.081	4.323	4.817	5.136	4.770	4.725
	TLR7	5.956	7.540	8.096	7.607	8.429	8.024	8.625	7.297
	TLR9	9.722	9.387	9.163	9.784	9.351	9.348	9.233	9.682
	TNFRSF10A	6.972	6.365	5.978	6.737	6.874	6.844	6.062	7.242
	TNFRSF10B	7.337	7.375	7.267	7.875	7.714	7.527	7.849	7.361
	TNFSF10	6.467	7.168	8.032	8.030	7.627	8.749	8.183	7.017
	TP53	7.162	6.983	7.354	7.519	7.199	7.335	7.130	7.110
	TRADD	6.105	5.322	5.821	6.266	5.964	6.077	5.538	5.576
	TRAF1	6.586	5.797	4.361	6.126	4.914	5.987	4.360	6.170
	TRAF2	7.484	6.803	6.578	7.069	6.804	6.854	6.330	7.113
	TRAF6	6.409	6.230	6.582	6.249	6.292	6.284	6.542	6.578
	TYK2	9.360	9.088	9.355	9.545	9.577	9.561	9.388	8.886
	XIAP	8.070	7.988	7.761	7.674	7.847	7.851	7.858	8.231

Table SA10 :

 SA10 Log-transformed gene expression for multiple sclerosis (MS) patients, in the 'alert' (M0) macrophage activation state.

	M1	MS 1	MS 2	MS 3	MS 4	MS 5	MS 6	MS 7	MS 8
	AIFM1	8.601	8.526	8.354	8.205	8.233	8.121	8.125	8.595
	APAF1	8.021	8.310	8.487	8.082	7.805	8.110	7.803	8.221
	ATF2	8.630	8.101	8.563	8.882	8.839	9.046	9.183	9.738
	BAK1	8.892	8.817	8.830	9.262	9.104	9.300	9.069	9.051
	BAX	8.396	8.780	8.968	9.103	9.105	9.021	9.199	8.924
	BBC3	7.612	7.603	7.742	8.519	8.134	8.020	8.159	8.772
	BCL2	5.089	5.237	5.846	6.099	5.553	5.860	5.879	6.532
	BCL2L1	8.538	8.532	8.698	8.729	8.634	8.575	8.385	10.140
	BID	8.695	8.986	9.362	8.963	9.029	8.627	8.781	9.042
	BIRC2	8.679	8.807	8.900	8.937	8.686	9.038	8.690	10.100
	BIRC3	10.206	10.020	10.034	10.364	10.175	10.546	10.181	11.783
	CASP1	10.015	10.171	10.258	10.429	9.754	10.317	10.220	10.422
	CASP10	7.968	8.328	8.216	8.222	7.625	7.712	7.717	7.792
	CASP2	7.702	7.709	7.694	7.519	7.406	7.428	7.829	7.125
	CASP3	8.298	8.411	8.066	8.346	7.486	8.176	7.673	7.790
	CASP4	10.421	10.016	10.362	10.868	10.476	10.936	10.984	10.868
	CASP6	6.511	6.434	6.020	6.630	6.448	6.672	6.466	6.633
	CASP7	8.499	8.224	8.502	9.273	8.297	9.034	8.918	9.666
	CASP8	7.334	7.541	7.501	7.405	7.317	7.348	7.349	7.395
	CASP9	7.122	7.289	7.105	7.198	7.250	7.037	7.058	7.238
	CCL5	11.961	11.362	12.029	13.159	12.973	13.426	13.135	14.435
	CFLAR	10.012	9.818	10.027	10.401	9.778	10.235	10.173	11.121
	CHUK	8.683	9.003	9.074	8.476	8.382	8.395	8.614	8.548
	CIITA	9.609	9.063	9.321	9.458	9.427	9.166	10.092	9.019
	CXCL10	12.600	13.330	13.587	13.983	11.805	13.315	13.661	12.497
	CXCL11	10.645	11.651	11.974	12.602	10.099	11.720	12.249	11.121
	CXCL9	12.982	13.714	14.126	13.580	13.077	13.890	14.763	13.770
	CYCS	9.747	10.347	10.326	9.482	9.366	9.256	9.333	9.843
	DFFA	7.319	6.910	6.814	7.241	7.130	7.304	7.403	7.112
	DIABLO	9.172	9.162	9.233	9.080	9.022	9.091	9.191	9.105
	EIF2AK2	10.112	9.712	9.640	10.553	9.707	10.302	10.114	10.654
	ENDOG	5.192	4.747	4.115	5.151	5.065	4.787	3.913	4.248
	EP300	9.708	9.311	9.646	9.632	9.697	9.709	9.791	10.183
	FADD	8.385	8.389	8.673	8.650	8.494	8.408	8.490	8.323
	FAS	7.142	8.048	8.614	8.398	7.956	8.422	8.584	9.120
	FASLG	1.083 9.639	9.240	9.086	9.802	9.298	9.719	9.570	9.806
	TBK1	9.589	9.464	9.566	9.587	9.442	9.558	9.649	10.053
	TICAM1	9.635	8.804	9.026	9.834	9.378	9.721	9.787	11.242
	TICAM2	9.260	9.352	9.568	9.321	9.061	9.329	9.475	9.356
	TLR3	6.061	5.750	5.530	6.802	6.700	6.754	5.368	5.410
	TLR7	7.173	8.131	7.692	8.189	6.819	7.727	7.296	8.015
	TLR9	9.612	9.561	9.728	9.632	9.833	9.398	9.277	8.955
	TNFRSF10A	8.090	7.737	7.591	8.432	8.120	8.589	7.864	8.572
	TNFRSF10B	6.938	7.558	8.135	8.056	8.155	7.963	8.111	8.056
	TNFSF10	11.269	11.146	11.274	12.170	11.233	11.893	11.412	11.157
	TP53	7.167	7.098	7.104	7.141	7.084	6.698	7.130	6.612
	TRADD	7.117	6.877	7.069	7.896	7.252	7.817	6.763	7.162
	TRAF1	8.603	8.977	9.088	9.643	8.712	9.565	9.301	11.014
	TRAF2	8.372	8.047	7.840	8.417	8.109	8.290	8.185	7.784
	TRAF6	7.062	6.932	7.021	7.321	6.927	7.357	7.373	7.845
	TYK2	9.644	9.532	9.349	9.676	9.958	9.558	9.487	9.269
	XIAP	8.161	8.219	8.124	8.003	8.079	8.353	8.026	9.043

Table SA11 :

 SA11 Log-transformed gene expression for multiple sclerosis (MS) patients, in the 'pro-inflammatory' (M1) macrophage activation state.

	Gene	Delta	Gene (cont.)	Delta (cont.)
	AIFM1	0.025	IL15	0.089
	APAF1	0.019	IL1B	1.261
	ATF2	0.067	IRF1	0.349
	BAK1	0.128	IRF2	-0.019
	BAX	0.030	IRF3	-0.076
	BBC3	0.243	IRF4	0.484
	BCL2	0.441	IRF5	-0.061
	BCL2L1	0.348	IRF7	0.192
	BID	0.454	IRF8	0.452
	BIRC2	0.188	IRF9	-0.046
	BIRC3	1.147	JAK1	0.139
	CASP1	0.053	JAK2	0.092
	CASP10	-0.126	MDM2	0.091
	CASP2	-0.233	MYD88	0.019
	CASP3	-0.116	NFKB1	0.451
	CASP4	0.018	NFKB2	0.687
	CASP6	-0.417	NFKBIA	0.698
	CASP7	0.127	NOS2	-0.073
	CASP8	-0.184	OAS1	0.043
	CASP9	0.052	PARP1	-0.140
	CCL5	1.068	PLSCR1	0.191
	CFLAR	0.327	PRKACA	0.016
	CHUK	0.105	PRKCD	0.172
	CIITA	-0.285	PRKRA	0.090
	CXCL10	2.324	PSMB9	0.048
	CXCL11	2.388	RELA	0.215
	CXCL9	1.670	RELB	0.710
	CYCS	0.308	RIPK1	-0.074
	DFFA	-0.126	SOCS1	0.647
	DIABLO -0.002	SOCS3	0.694
	EIF2AK2	0.021	SOD2	0.920
	ENDOG	0.084	SP100	-0.022
	EP300	0.025	SPI1	-0.017
	FADD	0.071	STAT1	0.210
	FAS	0.437	STAT2	0.162
	FASLG	0.286	TBK1	0.095
	GBP1	0.820	TICAM1	0.276
	HLA-B	0.119	TICAM2	0.019
	ICAM1	0.698	TLR3	-0.079
	IFNA1	-0.859	TLR7	-0.569
	IFNAR1	0.135	TLR9	0.076
	IFNAR2	-0.016 TNFRSF10A	0.226
	IFNB1	0.446	TNFRSF10B	0.296
	IFNG	0.688	TNFSF10	-0.270
	IFNGR1	0.027	TP53	-0.021
	IFNGR2	-0.017	TRADD	0.130
	IKBKB	-0.062	TRAF1	0.984
	IKBKE	0.326	TRAF2	0.274
	IKBKG	0.069	TRAF6	0.027
	IL12B	2.542	TYK2	-0.078
			XIAP	0.223

Table SA12 :

 SA12 Values of ∆ for each gene(5.1.6). The value of the threshold at the 75th percentile is 0.438.

(a) Indegree of the structural systems.(b) Outdegree of the structural systems.

Bibliography

by a linear time invariant dynamics. While it is known that linear models can provide useful information [START_REF] Gu | Controllability of structural brain networks[END_REF][START_REF] Liu | Control Principles of Complex Networks[END_REF], this kind of dynamics is not always appropriate for biological systems such as the brain [START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF]. However, our algorithm is flexible, and future developments may include different criteria to perform the controllability test, such as the Popov-Belevitch-Hautus criterion [START_REF] Camlibel | Popov-Belevitch-Hautus type controllability tests for linear complementarity systems[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF], or the Gramian condition [START_REF] Chen | Linear System Theory and Design, International 3rd. Edition[END_REF]. A different controllability test altogether, for instance a feedback vertex set [START_REF] Zañudo | Structure-based control of complex networks with nonlinear dynamics[END_REF] approach, able to deal with nonlinear dynamics, may be implemented.

A seemingly subjective choice in our algorithm is the choice of an ordering for the target set. Future work may involve the study of the effect of different orderings on standard graph models such as Erdös-Rényi random graphs [START_REF] Erdös | On the evolution of random graphs[END_REF] and Barabási-Albert scale free networks [START_REF] Barabási | Emergence of Scaling in Random Networks[END_REF], in order to inform the choice in a principled way.

To understand complex biological systems we require expertise from different fields, studying them from a control-theoretical perspective offers the promise to a new insight.

Scientific production

Submitted journal papers

• Bassignana G., Fransson J., Henry V., Colliot O., Zujovic V., De Vico Fallani F., Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis. 2020. arXiv:2003.08913v3. [START_REF] Bassignana | Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis[END_REF] • Gonzalez-Astudillo J., Cattai T., Bassignana G., Corsi M.-C., and De Vico Fallani F., Network-based brain computer interfaces: principles and applications. 2020. arXiv:2006.13187. [START_REF] Gonzalez-Astudillo | Network-based brain computer interfaces: Principles and applications[END_REF] • Ansart M., Epelbaum S., Bassignana G., Bône A., Bottani S., et al., Predicting the Progression of Mild Cognitive Impairment Using Machine Learning: A Systematic and Quantitative Review. 2019. hal-02337815. [START_REF] Ansart | Predicting the Progression of Mild Cognitive Impairment Using Machine Learning: A Systematic and Quantitative Review[END_REF] In preparation journal papers • Bassignana G., Colliot O., De Vico Fallani F., A step-wise criterion to determine target control centrality in the aging brain network.

• Henry V., Bassignana G., Lejeune F. X., Zujovic V., De Vico Fallani F., Dameron O., Moszer I., Colliot O., OWL ontology and SWRL rules applied to systems biology networks conciliation.

Conference talks and posters

• Poster presentation at Network neuroscience online conference and oral presentation at Network neuroscience satellite, September 2020.

• Oral presentation at Complex networks: from socio-economic systems to biology and brain workshop, Lipari, Italy, July 2019.

• Oral presentations at Network science (NetSci) conference and Network neuroscience satellite, Burlington, VT, May 2019.

• Oral presentation at IEEE workshop on complexity in engineering, Florence, Italy, October 2018.

• Poster presentation at Conference on Complex Systems, Thessaloniki, Greece, September 2018.

• Lightning talk and poster presentation at Network science (NetSci) conference, Paris, France, June 2018.

Conference attendance

• IEEE Conference on Decision and Control, Nice, France, December 2019.

Appendix A

Supplementary material for Section 5.1 A.1 Supplementary figures

B.2 Supplementary tables

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