Structural Factorization of Plants to Compute their Functional and Architectural Growth

Numerical simulation of plant growth has been facing a bottleneck due to the cumbersome computation implied by the complex plant topological structure. In this paper, we present a new mathematical model for plant growth, GreenLab, overcoming these difficulties. GreenLab is based on a powerful factorization of the plant structure. Fast simulation algorithms are derived for deterministic and stochastic trees. The computation time no longer depends on the number of organs and grows at most quadratically with the age of the plant. This factorization finds applications in the context of Geometric Models, to build trees very efficiently, and in the context of Functional Structural Models, to compute biomass production and distribution.

Introduction

Virtual plants are more and more used for different applications, mainly in computer graphics (urbanism and landscaping) and environmental sciences (agronomy and forestry). Sievänen in [START_REF] Sievänen | Components of a functional-structural tree model[END_REF] distinguishes three kinds of models for virtual plants: Process Based Models (PBM), Geometric Models (GM) and Functional Structural Models (FSM). Process Based Models focus on ecophysiological processes and are used to answer agronomic questions (see for example GOSSYM for cotton [START_REF] Baker | Gossym: a simulation of crop growth and yield[END_REF], CERES for Maize [START_REF] Jones | CERES -Maïze : A simulation model of maïze growth and development[END_REF] or [START_REF] Marcelis | Modelling of biomass production and yield of horticultural crops: a review[END_REF]] for a review). They typically describe the flux of external resources through a system (the plant or the crop) into a pool (the yield). A set of rules and parameters govern resource acquisition and conversion on a field area basis, according to a phenological timetable. This approach is particularly robust for field crops forming homogeneous canopies. It is no longer true for heterogeneous populations or culture systems in which plant architecture and geometry are of specific interest to the producer. The development and management of these systems require models providing the plant architectural description, in addition to resource flows and yield. Such models should also integrate the plant phenotypic plasticity resulting from feedbacks between functional growth and architectural development. Geometric Models refer to plant architectural models built from simulation algorithms for computer graphics needs, (for pioneering works, see [START_REF] Aono | Botanical tree image generation[END_REF], [START_REF] Reffye (de) | Plant models faithful to botanical structure and development[END_REF] or [START_REF] Prusinkiewicz | Algorithmic Beauty of Plants[END_REF]). They are only based on a topological description of the plant and some geometric rules. Handled by good designers, these models produce authentic looking pictures of plants. Recent interesting examples can be found in [START_REF] Deussen | Digital Design of Nature -computer generated plants and organics[END_REF]). If it works quite well for little plants (at least from a visual point of view), limitations occur so far when it comes to the representation of trees, as their botanical developments is too complex and strongly influenced by physiological functioning. Evolved software built on this type of model is commonly used in landscaping, advertising or video games but is of little interest for agro-ecological research, as no biomass is computed. Developed recently, Functional Structural Models are mainly devoted to agronomy and forestry. They attend to combine Process Based Models and Geometric Models in order to control both photosynthesis and organogenesis. It is generally done by performing a dynamic simulation of plant morphogenesis. The plant architectural development provides the sources according to which the photosynthesis is determined and the compartments among which the produced biomass is distributed, see for example LIGNUM model [START_REF] Perttunen | Lignum: a tree model based on simple structural units[END_REF] and AMAPhydro [Reffye (de) et al., 1997]. The need for fast simulation and visualization tools is obvious for computer graphics purposes. It is also crucial for agro-ecological research. As a matter of fact, for applications, plant growth models have to be calibrated on real plant measurements. Parameter identification can only be performed by running a very large number of times the growth simulation process. The same issue is raised when solving optimal control or optimization problems, see [Wu, 2005]. Most simulation algorithms rely on the complete construction of the plant topology by mimicking the simultaneous bud functioning, see for example [START_REF] Prusinkiewicz | Algorithmic Beauty of Plants[END_REF] and other related works based on L-systems or automata. For Geometric Models, Sievänen observed empirically that the computational time is proportional to the square of the number of items, see [START_REF] Sievänen | Components of a functional-structural tree model[END_REF]. When it comes to Functional Structural Models, the simulation of source and sink processes, biomass acquisition and distribution requires exploring several times the plant topological structure. It increases heavily the computational time (up to 8 times). In AMAPhydro model, the total biomass production is obtained by summing the photosynthetic production of every leaf and biomass partitioning is computed by determining the individual demand of all organs and then allocating biomass to each of them, see [START_REF] Blaise | Modeling of metamorphosis and spatial interactions in the architecture and development of plants[END_REF]. But for branching plants, the number of organs grows exponentially, which leads to cumbersome computations. As a consequence, the algorithms can be drastically improved by taking into account the fact that a plant is often composed of a juxtaposition of similar substructures. In this paper, we will present an improved method of plant simulation based on a powerful factorization of the plant architectural organization. This factorization method is applied to the functional structural model GreenLab detailed in [Reffye (de) and Hu, 2003] and [START_REF] Yan | A dynamic, architectural plant model simulating resource-dependent growth[END_REF], for deterministic and stochastic simulations. A real breakthrough is achieved as the bottleneck linked to complex tree architectures is suppressed. In section 2, we will present the main botanical and ecophysiological concepts used in the GreenLab model and build the dynamical system of plant growth. The factorization method is introduced in section 3. A new mathematical formalism is presented, which allows rewriting the equations of GreenLab and deriving a fast algorithm for plant growth simulation. In section 4, the advantages and drawbacks of the method are discussed, as well as the perspectives it opens for applications in both computer graphics and environmental sciences.

Basic Biological Concepts Used in the GreenLab Model

In this section, we present the botanical and ecophysiological concepts from which the GreenLab model was built. A consistent time unit for architectural growth and photosynthetic production is defined, which allows us to derive the discrete dynamical system of growth. A more detailed biological presentation of the GreenLab model can be found in [Reffye (de) and Hu, 2003] or [START_REF] Yan | A dynamic, architectural plant model simulating resource-dependent growth[END_REF].

Architectural Organization

As explained in [START_REF] Barthélémy | Architecture, gradients morphogénétiques et âge physiologique chez les végétaux[END_REF], the architecture of a plant can be seen as a hierarchical branching system in which the axes can be grouped into categories characterized by a particular combination of morphological parameters. Thus, the concept of physiological age was introduced to represent the different types of axes. For instance, on coffee trees, there are two types: orthotropic trunk and plagiotropic branches. Usually, we need less than 5 physiological ages to describe the axis typology in a given tree, see [START_REF] Sabatier | Growth dynamics and morphology of annual shoots according to their architectural position in young cedrs atlantica (endl.) manetti ex carrière (pinaceae)[END_REF] and the example of Cedrus Atlantica. The main trunk has a physiological age of 1 and the oldest physiological age is the ultimate state of differentiation for an axis, it is usually short, without branches. Moreover, botanists generally agree on the fact that plants are modular organisms that develop by the repetition of elementary botanical entities or constructional units, see for example [Bell, 1991] or [Barthélémy, 1991]. The most elementary of these entities is the metamer. It is composed of an internode bearing organs: buds, leaves, flowers. Depending on plants, metamers are set in place rhythmically or continuously. In the rhythmic case, see Figure 1, the plant grows by successive shoots of several metamers produced by buds. The apparition of these shoots defines the architectural Growth Cycle. A Growth Unit is the set of metamers built by a bud during a growth cycle. These metamers can be of different kinds and ordered according to botanical rules, like acrotony. To give a simple example, most temperate trees grow rhythmically, new shoots appearing at spring. If we do not consider polycyclism, their architectural growth cycle correspond to a year. Plant growth is said continuous when meristems keep on functioning and generate metamers one by one, see Figure 1. The number of metamers on a given axis (that is to say generated by the same meristem) is proportional to the sum of daily temperatures received by the plant. It is called the Law of the Sum of Temperatures in Agronomy, see [Jones, 1992] and [START_REF] Gao | Rice clock model -a computer model to simulate rice development[END_REF]. The Growth Cycle is defined as the thermal time unit necessary for a bud to build a new metamer. Tropical trees, bushes or agronomic plants have a continuous growth. In both continuous and rhythmic cases, the Chronological Age of a plant (or of an organ) is defined as the number of growth cycles it has been existing for.

Ecophysiological Concepts

GreenLab aims at providing a general framework for plant growth models and their applications to agronomy and forestry. For this purpose, it is necessary to add to the architectural model some ecophysiological submodels to compute the photosynthetic biomass production and its distribution among organs. Here, we present the simple hypotheses used in GreenLab to describe plant functioning. However, it is important to note that more complex models could be implemented in the same framework, see [START_REF] Yan | A dynamic, architectural plant model simulating resource-dependent growth[END_REF]. We determine the photosynthetic biomass production by the law of Water Use Efficiency, see [START_REF] Howell | Relationship of dry matter production of field crops to water consumtion[END_REF]: the plant dry matter acquisition is approximately proportional to the plant transpiration. To be able to build the plant architecture, we need a notion of volume and weight of the wet plant biomass. For this purpose, in GreenLab, we convert the dry matter production into an available fresh biomass by assuming that the moisture content is constant. Therefore, in GreenLab, the water use efficiency will be considered as the conversion rate of water transpiration into fresh biomass. The fresh biomass thus computed is used to build organs. Each leaf will have a transpiration proportional to a potential transpiration rate and to its conductance depending on its surface area, see [Jones, 1992]. The Growth Cycle is deduced in both cases: it is defined as the time unit necessary for a new metamer to be set in place when the growth is continuous, or the period at which the Growth Units (GU) appear when the growth is rhythmic. For the sake of simplicity, metamers are drawn here without lateral buds.

Moreover, we do not consider any local diffusion process and make the hypothesis that the biomass produced by each leaf is stored in a common pool of reserves and redistributed among all organs according to their sink strengths. The initial seed and the leaves are sources. Leaves, internodes, fruits and rings (resulting from the plant secondary growth) are sinks. The root system is not described but is considered as a big sink at all stages of the plant growth. Secondary growth is the process controlling the increase of branch diameters. To describe it, we use the classical pipe model [START_REF] Shinozaki | A quantitative analysis of plant form -the pipe model theory i. basic analysis[END_REF]: the ring volume of a branch at a given spot in the plant architecture is proportional to the number of leaves seen above.

GreenLab Growth Cycle

The time unit for organogenesis and photosynthesis must be consistent, insuring a good synchronization between the two processes. In GreenLab, we consider that the new growth units appear at the beginning of the growth cycle and that, during this one, the plant architecture remains constant. As illustrated by Figure 1, it is a good approximation for rhythmic growth. In the case of continuous growth, it corresponds to a discretization of the system with respect to the growth cycle. Growth units are thus simply composed of one metamer. The discretization is reasonable in the continuous case as the cycle duration is usually quite short compared to that of the whole growth process. Consequently, we will no longer distinguish rhythmic and continuous growths.

We suppose as well that at the level of growth cycle, the fluctuations of water use efficiency and of potential transpiration rate are reasonable, and we use their average values at growth cycle n. Also knowing the architecture at growth cycle n, and thus leaf surface areas, we can determine their transpiration and the total amount of produced biomass, denoted Q(n). At the end of the cycle, fresh matter is allocated to organs according to their relative demands (sinks): old organs in expansion, new organs preformed in buds (they will appear at the beginning of growth cycle n + 1), root system, rings... Empirical allometric rules are used to determine the dimensions and shapes of organs from their volumes, and the plant architecture at growth cycle n + 1 is thus obtained. Figure 2 illustrates the different steps to compute the plant functioning during its growth cycle. From this flowchart is derived the algorithms to simulate plant growth and development.

Figure 2: Plant functioning flowchart: the seed gives the initial pool of biomass. It is used to build organs (internodes, roots, leaves, fruits) and thus the plant architecture. Roots allow water assimilation used for leaf transpiration and photosynthetic biomass production. This biomass is stored in the common pool of reserves and is then distributed among organs, which ends the growth cycle.

A New Way to Simulate Plant Growth Based on its Structural Factorization

It is possible to simulate tree development and growth thanks to computer simulation techniques that imitate closely processes of plant functioning, i.e. handle the growth of each organ individually. Every bud organogenetic production is simulated, which means that all organs are created one by one. Likewise, the production and the circulation of fresh biomass use the paths of the topological and geometrical structures in the tree architecture, all organs have to be reached. Such simulation process was implemented in AMAPhydro Software, see [START_REF] Reffye (de) | A functional model of tree growth and tree architecture[END_REF] and [START_REF] Blaise | Modeling of metamorphosis and spatial interactions in the architecture and development of plants[END_REF]. But this method leads to cumbersome computations.

The factorization method we present here strongly relies on the plant botanical organization. It takes into account the fact that identical organs and structures are repeated a large number of times in the plant architecture. We derive an efficient method of simulation based on plant instantiations, instead of simulating all organs.

Organogenesis

Botanical Instantiations and Notations

Metamers and buds: Let P be the maximum number of physiological ages in the plant.

It is generally very small, inferior to 5, [START_REF] Sabatier | Growth dynamics and morphology of annual shoots according to their architectural position in young cedrs atlantica (endl.) manetti ex carrière (pinaceae)[END_REF]. From the botanical description of the plant, we get that metamers and buds are the elementary bricks of the plant structure.

At growth cycle t, a metamer is characterized by its physiological age p, the physiological age of its axillary branches q (with q ≥ p; if there is no axillary branch, we take q = P + 1) and its chronological age n. It will be denoted by m t pq (n) or simply m pq (n) if there is no ambiguity on the growth cycle. These 3 indices p, q, n are sufficient to describe all the metamers composing a plant of chronological age t and we have at most 1/2 P (P + 3)t different metamers. A bud is only characterized by its physiological age p and will be denoted by s t p or s p . We have of course at most P different buds in a plant.

Substructures:

The terminal bud of a plant axis produces different kinds of metamers bearing axillary buds of various physiological ages. These buds themselves give birth to axillary branches and so on... A substructure is the complete plant structure that is generated after one or several cycles by a bud. In the deterministic case, all the substructures with the same physiologicaland chornological ages are identical if they have been set in place at the same moment in the tree architecture. At cycle t, a substructure is thus characterized by its physiological age p and its chronological age n. It will be denoted by S t p (n). We recall here that the physiological age of the main trunk is 1. Thus, at growth cycle t, the substructure of physiological age 1 and of chronological age t, S t 1 (t), represents the whole plant. Figure 3 illustrates the way substructures are organized. The total number of different substructures in a plant of chronological age t is thus P t. This number is very small, usually less than 30, even if the total number of organs is high. Substructures and metamers will be repeated a lot of times in the tree architecture, but they need to be computed only once for each kind.

Mathematical Formalism

This part is based on [Reffye (de) et al., 2003]. We develop and adapt their formalism. Without loss of generality and in order to simplify the notations, we will suppose that all metamers bear axillary buds and we will no longer consider the virtual physiological age P + 1. We will also suppose that there is no mutation (the apical bud does not change its physiological age) while the axis grows. Let G t be the set of metamers and buds:

G t = m t pq (n), 1 ≤ p ≤ P, p ≤ q ≤ P, 1 ≤ n ≤ t ∪ s t p , 1 ≤ p ≤ P . (1)
We consider the monoid S t generated by G t for the concatenation internal operator. This operator will be represented by the multiplicative sign. Substructures are thus elements of S t . They are built by induction, as follows:

• Substructures of chronological age 0 are buds:

S t p (0) = s t p , (2)
Figure 3: Substructures of physiological ages 1, 2, 3 at chronological ages 0, 1, 2 and their organization: S 1 (0), S 2 (0), S 3 (0) are buds of physiological age 1, 2, 3 respectively. In this example, a growth unit of physiological age 1 is composed of 2 metamers of type m 12 and 1 metamer of type m 13 ; a growth unit of physiological age 2 is composed of 2 metamers of type m 23 ; a growth unit of physiological age 3 is composed of 1 metamer of type m 34 (without axillary bud).

• and if we suppose built all the substructures of chronological age n -1, we deduce the substructures of chronological age n:

S t p (n) =   p≤q≤P m t pq (n) upq(t+1-n) S t q (n -1) bpq(t+1-n)   S t p (n -1) . (3)
For all (p, q) (such that 1 ≤ p ≤ P, p ≤ q ≤ P ), (u pq (t)) t and (b pq (t)) t are sequences of integers that are characteristic of the plant organogenesis: u pq (t) corresponds to the number of metamers m pq in growth units of physiological age p appearing at growth cycle t; b pq (t) is the number of axillary substructures of physiological age q in growth units of physiological age p that appeared at growth cycle t.

In equation ( 3), substructure S t p (n) is decomposed into : -its oldest growth unit, called base growth unit:

p≤q≤P m t pq (n) upq(t+1-n)
-the lateral substructures borne by the base growth unit (they are one cycle younger):

p≤q≤P S t q (n -1) bpq(t+1-n)
-the substructure grown from the apical bud of the base growth unit (also one cycle younger):

S t p (n -1)
This decomposition is illustrated on S 1 (2) in Figure 3.

The GreenLab organogenesis model has been derived in 3 forms:

• GL1 corresponds to the deterministic organogenesis model, without influence of the plant functioning. Mathematically, it corresponds to u pq and b pq constant, see [START_REF] Yan | A dynamic, architectural plant model simulating resource-dependent growth[END_REF].

• GL2 corresponds to a stochastic model of organogenesis, u pq and b pq are stochastic variables. As a consequence, substructures of the same chronological and physiological ages can be very different: S t p (n) is a stochastic variable with values in S t . More details are given in section 3.1.4 and in [START_REF] Kang | Theoretical study and numerical simulation of a stochastic model for plant growth[END_REF].

• GL3 corresponds to a deterministic model with total retroaction between organogenesis and photosynthesis. u pq (t) and b pq (t) are functions of the biomass produced by the plant at growth cycle t -1 and t respectively, see [START_REF] Mathieu | A dynamical model of plant growth with full retroaction between organogenesis and photosynthesis[END_REF].

From equation (3), we can derive the algorithms to build the topological structure of the plant at any growth cycle t, in both deterministic and stochastic cases. The numbers of organs can also be computed as a result of organogenesis. These numbers will prove of crucial importance in section 3.2 to compute plant functioning.

Deterministic plants

Topology: The inductive construction of section 3.1.2 gives the algorithm to build the topological structure of the plant at any growth cycle t. We suppose that the sequences (u pq (t)) t and (b pq (t)) t are known. To get the full plant structure at cycle t, we need to build all the substructures S t p (n), for all p ∈ [1; P ] and for all n ∈ [0; t]. It is simply done using equation (3) and by induction on n, as illustrated by Figure 3. If (u pq (t)) t and (b pq (t)) t are constant (GL1 case), S t p (n) are (topologically) independent of t, and the construction does not have to be done at each growth cycle but only once.

Numbers of Organs: Without factorization of the plant structure, counting the number of organs is a typical bottleneck; the computing time can be tremendous for big trees and forests. Here, we overcome this difficulty. In order to determine the number of organs, we count the metamers. For this purpose, we define the morphism Ψ of S t into (IR P , +). If S is in S t , Ψ(S) is the vector of the number of metamers of each physiological age in S. On the generating set G t :

Ψ(m t pq (n)) = I p and Ψ(s t p ) = (0) (4)
where:

I p =             0 . . . 0 1 0 . . . 0             ← rank p (5)
Then we deduce Ψ(S t p (n) for all p, n using the morphism property and by induction on n. From equation (3):

Ψ S t p (n) = Ψ     p≤q≤P m t pq (n) upq(t+1-n) S t q (n -1) bpq(t+1-n)   S t p (n -1)   = p≤q≤P u pq (t+1-n) I p + p≤q≤P b pq (t+1-n)Ψ S t q (n -1) + Ψ S t p (n -1)
(6) An interesting result can be derived for the GL1 model, that is to say when u pq and b pq are constant, and if there is no mutation. In this case, Ψ(S t p (n)) is independent of t. Thus, if we introduce the square matrix of order P :

U n = (Ψ(S 1 (n)), • • • , Ψ(S P (n))) , (7)
we obtain the following system:

U n = V + U n-1 (I + N ) , ( 8
)
where I is the identity matrix of order P , V is a diagonal matrix, V ij = δ ij i≤q≤P u iq and N is an inferior triangular matrix, N ij = b ji . We obtain:

U n = V n-1 i=0 (I + N ) i . (9)
Without reiteration (if b pp = 0 for all p), N is nilpotent and the equation simplifies.

This inductive plant construction is very fast. The computation time is proportional to P t 2 (P t for GL1) and not to the number of metamers. As the number of organs (internodes, leaves, fruits...) per metamer is botanically known, GreenLab provides a mathematical tool that enables to compute the organ production of a virtual plant very quickly and thus suppresses the drawback of counting the organs one by one by simulation.

Geometric Representation: If we add geometric rules (internode lengths, branching angles, phyllotaxy) to the construction algorithm, we will obtain the 3D architecture of a geometrical tree, [Reffye (de) et al., 2003]. We can extend the role of substructures to be sets of polygons storing their geometric shapes into a library. They become meta-organs, and positioning a substructure in the plant architecture needs the same operations as for a simple 3D organ. They can also be displayed separately, which can be useful. The tree of Figure 3.1.3 built with AMAPsim software can be computed either with the classical method simulating every organ or with substructure factorization, see [START_REF] Yan | Fast construction of plant architectural models based on substructure decomposition[END_REF]. Resulting computational times are compared: more than one hour for the total simulation of the 15 year old plant, less than one second with the substructure factorization! The mathematical algorithm proposed totally suppresses the simulation cost of tree architecture.

Stochastic plants -GL2 model

Topology: The inductive construction algorithm can be extended to stochastic plants, when (u pq (t)) t and (b pq (t)) t are stochastic variables controlling growth, branching and death processes, see [START_REF] Kang | Theoretical study and numerical simulation of a stochastic model for plant growth[END_REF] adapted as follows:

S t p (n) = p≤q≤P m t pq (n) upq(t+1-n) 1≤i≤bpq(t+1-n) S t q (n -1) i S t p (n -1) , ( 10
)
where S t p (n) denotes a realization of the stochastic variable S t p (n). In the case of deterministic growth, for given chronological and physiological ages, there is only one substructure. For the stochastic growth, all the realizations of S t p (n) could be different. As a consequence, we lose the advantage of the substructure decomposition as no factorization is possible. For simulation, we overcome this difficulty by taking a limited number of samples of substructures that are realizations of the same stochastic distribution. These samples of substructures in a given set (at growth cycle t, for given physiological and chronological ages p and n) are obtained from Equation ( 10): for each sample, we get values for (u pq (t)) t and (b pq (t)) t and choose S t q (n -1) in the appropriate sets. The same loop as in the deterministic case is used, which insures that these sets have already been built. On Figure 5, we give examples of these sampling sets and how samples are used in structures of higher chronological ages.

The size R of the substructure sets can vary. Of course, the accuracy will increase with R. This method to simulate stochastic trees is very efficient. The time necessary to build the first tree is proportional to P × t 2 × R. Once again, it does not depend on the number of organs to create. It is important to note that the algorithm is even more interesting to build stochastic forests, as the sets of substructures built for the first stochastic tree can be used to build other trees of the same kind that are realizations of the same stochastic distribution.

Numbers of Organs: Similarly to the deterministic case, the numbers of organs can be deduced from the number of metamers. We will thus compute its stochastic distribution. Without bud death processes, the metamer distribution results from parallel stochastic processes and it is very close to a normal distribution. So the mean and variance are sufficient to obtain the distribution shape with accuracy. Using compound law formulas (see [START_REF] Sedgewick | Introduction to Algorithm Analysis[END_REF]), it is possible to compute the theoretical mean, variance and covariances of the number of metamers in a plant whose topological structure is given by Equation ( 10). The theoretical formulas avoid heavy Monte-Carlo simulations to get the distribution shape as it used to be done with several days of computation! It is also possible to get a very good approximation of the mean and variance of the produced biomass using differential statistics applied to the production equation. The detailed calculations and formulas are given by [START_REF] Kang | Theoretical study and numerical simulation of a stochastic model for plant growth[END_REF]. These results are synthetic and do not need plant reconstruction. For many experiments they can be quite sufficient to assess the mean and variance of the yield.

Plant Functioning

The plant structural factorization given in Equation ( 3) also allows us a fast computation of the plant functioning. As explained in section 2.2, we consider two steps at each growth cycle: first, the biomass is produced by leaf photosynthesis and stored in a common pool, and then, it is redistributed among all organs according to their sink strengths. Both steps strongly rely on the numbers of organs.

The Operator of Growth

The growth of a plant from cycle t to cycle t + 1 is given by the growth of all its substructures. We define the evolution operator φ t of S t into S t+1 as a morphism for the concatenation law. As G t is a generating set of S t , the morphism φ t is fully determined by its values on G t . We have:

           φ t (m t pq (n)) = m t+1 pq (n + 1) , with 1 ≤ p ≤ P, p ≤ q ≤ P, 1 ≤ n ≤ t φ t (s t p ) =   p≤q≤P m t+1 pq (1) upq(t+1) s t+1 q bpq(t+1)   s t+1 p , with 1 ≤ p ≤ P (11)
The initial condition is given by s 0 1 , which is the seed. If we only consider organogenesis without functioning, we have m t pq (n) = m pq for all t, n, and this formalism is equivalent to the dual-scale automaton, see [Hu et al., 2003], and to grammars or L-Systems, see [START_REF] Prusinkiewicz | Algorithmic Beauty of Plants[END_REF]. However, in their original forms these formalisms can not take properly into account functioning. A notable improvement on this regard was the development of sensitive growth grammars, see [Kurth, 1994] and [START_REF] Kurth | Growth grammars simulating trees-an extension of l-systems incorporating local variables and sensitivity[END_REF], attaching local variables to plant organs. In a similar way, substructures can integrate all necessary variables for plant functioning, but the main advantage is that the plant factorization allows a very quick treatment of these variables.

Biomass Acquisition -Photosynthesis

Every green leaf i produces fresh biomass that fills the pool of reserves. Its production is given by the water use efficiency times its water transpiration, and the leaf transpiration is obtained by the potential transpiration rate multiplied by the leaf conductance, see [Jones, 1992] or [START_REF] Allen | Crop evapotranspiration. guidelines for computing crop water requirements[END_REF]]. E t denotes the water use efficiency times the potential transpiration rate at growth cycle t and sums up all the environmental effects. In its extremely simple version presented here, the model applies a constant E t to all leaves. An empirical nonlinear function f depending on the leaf surface A i and on physiological parameters r 1 , r 2 gives the leaf conductance. The plant biomass production is thus:

Q t = E t i f (A i , r 1 , r 2 ) , (12)
The empirical function f chosen in GreenLab is:

f (A i , r 1 , r 2 ) = 1 r 1 /A i + r 2 . ( 13
)
This function can be easily changed according to modellers' choices.

We define the conductance of a metamer as the sum of conductances of all its leaves and we introduce the morphism Γ of S t into (IR, +) on the generating set G t : Γ(m t pq (n)) gives the conductance of the metamer and Γ(s t p ) = 0. Then, we can deduce Γ(S t p (n)) by induction on n. First, we know that for all p, 1 ≤ p ≤ P , Γ(S t p (0)) = Γ(s t p ) = 0. If Γ(S t p (n -1)) are known for all p, we use equation 3:

Γ S t p (n) = Γ     p≤q≤P m t pq (n) upq(t+1-n) S t q (n -1) bpq(t+1-n)   S t p (n -1)   = p≤q≤P u pq (t+1-n)Γ m t pq (n) + b pq (t+1-n)Γ S t q (n -1) + Γ S t p (n -1)
(14) Finally, as the structure S t 1 (t) corresponds to the full plant, the production of biomass at growth cycle t is:

Q t = E t Γ S t 1 (t) . (15)

Biomass Distribution

Each organ o has a potential biomass attraction value that we name sink or organ demand. This sink p o depends on the type of organ, its physiological age and its chronological age n. n = 0 corresponds to organs that will appear at the next cycle (generated by the structures S t p (0), cf. Section 3.2.1), and n ≥ 1 to organs in expansion. The shape chosen for sink functions are left to the user's choice, but they should be able to fit properly any kind of sink variations, as a function of the organ chronological age. It must be flexible enough to give bell shapes, c or s shapes, etc.. More details on sink variation functions are given in [Reffye (de) and Hu, 2003]. We define the plant demand at growth cycle t as the total biomass attraction of all organs (leaves, internodes, fruits, layers, roots...):

D t = o p o , (16)
As for the computation of the numbers of organs or the biomass production, we introduce the morphism Λ of S t into (IR, +) and we define it on the generating set G t : Λ(m t pq (n)) gives the metamer demand, that is to say the sum of the sinks of all its organs for their expansion and secondary growth, and Λ(s t p ) is the demand of the growth unit that will be generated by the bud. We deduce Λ(S t p (n)) at every growth cycle t, for all p, n, by induction on n and we have:

D t = Λ S t 1 (t) (17)
The biomass q t o allocated to organ o at growth cycle t is thus:

q t o = p o Q t D t . (18)
Eventually, the organ volume depends on its apparent density and its dimensions on allometric rules. All this features can be measured directly from the organ shape. Particularly, we compute leaf surface areas that are used to determine photosynthesis at the next growth cycle.

Retroaction of Photosynthesis on Organogenesis

From Equation (18), we see the importance of the ratio Q t D t . It is the main regulating variable of plant growth. According to the environmental conditions and to the available biomass, the plant will adapt its topology along its growth. This phenomenon is taken into account in GL3, see [START_REF] Mathieu | A dynamical model of plant growth with full retroaction between organogenesis and photosynthesis[END_REF] or [Mathieu, 2006] for an exhaustive study. Regarding the topological equation (3), it corresponds to non constant sequences (u pq (t)) t and (b pq (t)) t :

u pq (t) = u pq Q t-1 D t-1 and b pq (t) = b pq Q t D t . ( 19
)
On Figure 6 appears very clearly the difference between GL1 and GL3. We simulate with DigiPlant software the growth of a tree of chronological age 15 in two different environmental conditions, E t = 0.5 and E t = 1. for all t. The two trees on the left are obtained with GL1: the plant keeps the same topology but is shorter in bad conditions. On the contrary, the two trees on the right are built with GL3: the plant changes topology with high plasticity and adapts to its environment. Concerning computational times, it is faster to compute GL1 topology than GL3 one (P × t structures to compute compared to P × t 2 ), but there is no difference concerning the computation of functioning and of organ dimensions. To give an idea, the computation time of the full GL3 tree with secondary growth (rings) is less than 0.1 second on standard PC.

Discussion

We proposed a topological factorization of plant architecture based on botanical instanciations, namely metamers and substructures. A proper mathematical formalism was adapted to this factorization and helped us derive a fast algorithm to simulate plant organogenesis (architectural development) in both deterministic and stochastic cases. Moreover, under the hypothesis of a homogeneous potential transpiration rate, we showed that the substructure factorization provides a very efficient framework to compute plant functioning (biomass production and distribution) when using ecophysiological submodels. The computational times varies at most quadratically with the plant chronological age. It would grow exponentially without structural factorization as the complexity would increase with the total number of organs.

The main drawback of the factorization method lies in the hypothesis that at a given growth cycle, all the substructures with the same physiological and chronological ages are identical and have the same functioning. It is too restrictive in numerous cases.

Firstly, the potential transpiration rate depends on the microclimatic conditions (temperature, light, humidity) perceived by each individual leaf. These conditions may vary significantly in the same plant. An average approach based on a homogeneous potential transpiration rate is generally sufficient to describe the biomass acquisition and distribution among organs, see [START_REF] Allen | Crop evapotranspiration. guidelines for computing crop water requirements[END_REF] or [START_REF] Guo | Parameter optimization and field validation of the functionalstructural model greenlab for maize[END_REF]]. However, it does not allow to take into account local effects which play a very important role for some applications, for example to study the development of pests (pathogens, insects), see [Chelle, 2005]. Another example of limitation concerns the biomechanical analysis of trees. An incremental finite element method was implemented in [START_REF] Fourcaud | Numerical modelling of shape regulation and growth stresses in trees ii: Implementation in the amappara software and simulation of tree growth[END_REF], based on an exhaustive simulation of the growth of all organs. The complete computation is thus very heavy. Such analysis can not be directly adapted to a simulation based on substructure factorization as each substructure repetition has its own spatial position and their own gravitational constraints.

In both cases, an adaptation to our method would consist in discretizing the array of varying conditions (value of potential transpiration rate in the first example, orientation of the substructure main axis in the second one). As in the stochastic organogenesis case, see section 3.1.4, we overcome the difficulty by handling a limited number of samples of substructures for given physiological and chronological ages, corresponding to the discretization. Fast plant growth simulation is very useful for graphical applications in urbanism or landscaping. Moreover, if plant geometric models are enriched with ecophysiological submodels, the visual impact is straightforward as plants look more authentic. However, we think that the most promising applications of substructure factorization concerns agronomy and forestry. To derive prediction and optimization tools, it is first necessary to identify model parameters using data obtained by experiments made on real plants. Whatever the identification algorithm used (newtoninan method with numerical computation of the system partial derivatives or heuristic methods, see [START_REF] Zhan | Fitting a structural-functional model with plant architectural data[END_REF]) the growth simulation process has to be run a very large number of times. The faster it is, the more efficient is the identification, particularly as speed allows a good interaction with the user. Moreover, sophisticated ecophysiological models can be implemented in the framework we proposed to simulate complex biological phenomena. Currently, the validation of the GreenLab model is widely studied. Several agronomic plants have already been calibrated (see the example of Maize in [START_REF] Guo | Parameter optimization and field validation of the functionalstructural model greenlab for maize[END_REF] or of branching plants like coffee trees or young beech trees in [Mathieu, 2006]). With the same applicative purpose, optimal control techniques are now applied to the model in order to optimize water or nutriant supplies, see [Wu, 2005]. Finally, the substructure factorization open interesting prospects as GreenLab remains consistent when changing scales. The complete model can be simplified without major drawbacks. Instead of creating compartments like process based models, we transform branches into metaorgans (i.e. substructures) being at the same time sources and sinks, whose values depend on the cumulative actions of the underlying organs. The ultimate transformation is to consider the tree as a single structure that is source and sink. The changing of scales relies on the equations of the full model and can be properly written.

Figure 1 :

 1 Figure1: The axis length is given as a function of time for continuous and rhythmic growths. The Growth Cycle is deduced in both cases: it is defined as the time unit necessary for a new metamer to be set in place when the growth is continuous, or the period at which the Growth Units (GU) appear when the growth is rhythmic. For the sake of simplicity, metamers are drawn here without lateral buds.

Figure 4 :

 4 Figure 4: Comparison of computing times (in seconds) between the simulation based on substructure factorization and the classical method simulating every organ (total simulation) at different growth cycles -Visualization of the plant at growth cycles 5 and 15

Figure 5 :

 5 Figure 5: Sampling sets and Stochastic Construction: for example, to build the set of S 3 (3) each time we need a substructure of type S 2 (2), we draw a sample in the corresponding set previoulsy built.

Figure 6 :

 6 Figure6: Effects of climatic variations on the GL1 and GL3 organogenesis models: on the left side (GL1), topology is fixed, organs are simply smaller to adjust to the environment; on the right side (GL3), toplogy is variable and the plant adapts its development to the environmental conditions.

Acknowledgement:

This work is supported in part by LIAMA (Sino-French Laboratory in Information, Automation and Applied Mathematics), Natural Science Foundation of China (#60073007), and China 863 Program (#2002AA241221).


Mark Kibanov
email: kibanov@cs.uni-kassel.de

Martin Atzmueller
email: atzmueller@cs.uni-kassel.de

Jens Illig
email: illig@cs.uni-kassel.de

Christoph Scholz
email: scholz@cs.uni-kassel.de

Alain Barrat
email: alain.barrat@cpt.univ-mrs.fr

Ciro Cattuto
email: ciro.cattuto@isi.it

Gerd Stumme
email: stumme@cs.uni-kassel.de

Is Web Content a Good Proxy for Real-Life Interaction? A Case Study Considering Online and Offline Interactions of Computer Scientists

Today, many people spend a lot of time online. Their social interactions captured in online social networks are an important part of the overall personal social profile, in addition to interactions taking place offline. This paper investigates whether relations captured by online social networks can be used as a proxy for the relations in offline social networks, such as networks of human face-to-face (F2F) proximity and coauthorship networks. Particularly, the paper focuses on interactions of computer scientists in online settings (homepages, social networks profiles and connections) and offline settings (scientific collaboration, faceto-face communications during the conferences). We focus on quantitative studies and investigate the structural similarities and correlations of the induced networks; in addition, we analyze implications between networks. Finally, we provide a qualitative user analysis to find characteristics of good and bad proxies.

I. INTRODUCTION

With the success of the Web and the rise of social media, many people spend a significant fraction of their time online. This is particularly the case in academia, as online activities can both correspond to work-related and private social networking. Online social relationship can then represent a significant share of an individual's social profile, in addition to the interactions that take place offline, e. g., when meeting friends, during a face-to-face conversation, etc. So far, the analysis of online social networks has received significant attention in the research community, while studies on offline interactions at large scale, e. g., focusing on networks of faceto-face proximity, has only been taken up recently. Studying human behavior using social media comes with some caveats [START_REF] Ruths | Social media for large studies of behavior[END_REF]: e. g., the responses of social media users differ from public opinion measured by surveys [START_REF] Mitchell | Twitter reaction to events often at odds with overall public opinion[END_REF]. Therefore, detailed investigations of the connection between individuals' offline and online interactions are needed. However, to the best of the authors' knowledge, a combined analysis focusing on inter-relations and implications between these different information sources has not been addressed before.

This paper aims to start filling this gap by analyzing online and offline social networks and published content of computer scientists who visited particular conferences. In particular, we investigate whether information from various online sources can be used as a proxy for the offline world. In order to answer this question, we proceed in three steps focusing on structural similarity, correlation, and implications of offline and online data. Basically, we investigate data structures and central parameters that are important for many applications, for example, for finding central actors, opportunistic forwarding, and for generating recommendations. For conducting our analysis and experiments, we investigate different online and offline datasets concerning computer scientists: we collected face-to-face interactions that took place during several computer science conferences, i. e., at LWA 2010 [START_REF] Atzmueller | Proceedings of the LWA 2010 -Lernen, Wissen, Adaptivität, Technical report (KIS)[END_REF], LWA 2011 [START_REF]Proceedings of the LWA 2011 -Lernen[END_REF], LWA 2012 [START_REF]Proceedings of the LWA 2012 -Lernen, Wissen, Adaptivität[END_REF] -organized by the german computer science society -and during ACM Hypertext 2011 [START_REF] Bra | HT'11[END_REF]. Specifically, by providing the CONFERATOR system [?] at these conferences we obtained offline interactions given by networks of human face-to-face proximity using RFID-technology developed by the SocioPatterns consortium. We also utilized a number of online resources and collected further data about the participants of these conferences: the content (full texts, abstracts, and titles) of their papers, the coauthorship networks based on DBLP information, their Research Gate profiles, and finally content from their homepages. The presented online and offline data and networks are not completely aligned by nature which makes the presented problem even more challenging and interesting.

For determining if (and when) online content is a good proxy for the offline world, we focus on different aspects and aim to answer the following research questions:

1) Are there strong structural similarities and correlations between the considered online and offline social networks w.r.t. link structure? (Q1) 2) Can we identify implications between online and offline networks and content? (i. e., concerning link structure and link importance, rankings, and (textual) content) (Q2) 3) Can we characterize classes of actors for which their online connections act as good proxies for their offline interactions, i. e., derive specific conditions using network parameters and properties of these actors? (Q3)

Our results indicate that online and offline datasets show some proxy relations but there are also specific structural differences.

Specifically, strong ties seem to correlate more than weak ones and "important persons" tend to keep their high rank from the online to the offline worlds. However, these correlations are often not strong enough for conclusions on the offline world using online data in all cases.

The rest of the paper is structured as follows. Section II discusses related work. Then, we introduce basic definitions and concepts in Section III. After that, we describe our datasets in Section IV. Section V discusses the results of our experiments. Finally, Section VI concludes the paper with a summary and promising options for future work.

II. RELATED WORK

The analysis of online and offline information, e. g., online social networks, co-authorship networks, co-location and networks of human face-to-face proximity is an interesting and challenging task in social network analysis and mining. Concerning human face-to-face contacts, the SocioPatterns collaboration developed an infrastructure that detects closerange and face-to-face proximity of individuals wearing RFID tags. This infrastructure has been deployed in various environments for studying the dynamics of human contacts, e. g., at conferences [START_REF] Atzmueller | Faceto-Face Contacts at a Conference: Dynamics of Communities and Roles[END_REF], [START_REF] Cattuto | Dynamics of Person-to-Person Interactions from Distributed RFID Sensor Networks[END_REF], [START_REF] Macek | Anatomy of a Conference[END_REF], in schools [START_REF] Pachucki | Mental Health and Social Networks in Early Adolescence: A Dynamic Study of Objectively-Measured Social Interaction Behaviors[END_REF], [START_REF] Stehlé | High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School[END_REF], museums [START_REF] Isella | What's in a Crowd? Analysis of Face-to-Face Behavioral Networks[END_REF] and workplaces [START_REF] Brown | Tracking Serendipitous Interactions: How Individual Cultures Shape the Office[END_REF], [START_REF] Sailer | Human versus Machine -Testing Validity and Insights of Manual and Automated Data Gathering Methods in Complex Buildings[END_REF]. [START_REF] Atzmueller | Faceto-Face Contacts at a Conference: Dynamics of Communities and Roles[END_REF] analyzes the interactions and dynamics of the behavior of participants at conferences; the connection between research interests, roles and academic jobs of conference attendees is further analyzed in [START_REF] Macek | Anatomy of a Conference[END_REF]. We extend these analysis focusing on offline and online information. Link relations and their prediction using different datasets (particularly, face-to-face contacts and DBLP data) is further analyzed in [START_REF] Scholz | New Insights and Methods For Predicting Face-To-Face Contacts[END_REF]. Also, [START_REF] Brown | Tracking Serendipitous Interactions: How Individual Cultures Shape the Office[END_REF], [START_REF] Sailer | Human versus Machine -Testing Validity and Insights of Manual and Automated Data Gathering Methods in Complex Buildings[END_REF], [START_REF] Smieszek | How should social mixing be measured: comparing web-based survey and sensor-based methods[END_REF] analyze survey data and networks of face-to-face proximity. A first analysis of online and offline networks has been performed in [START_REF] Barrat | Social Dynamics in Conferences: Analyses of Data from the Live Social Semantics Application[END_REF], however, focusing on the direction from offline to online data. In contrast, we focus on the reverse -investigating the use of online data as a proxy for offline interactions. Moreover, we extend the previous research made using face-to-face proximity data and discuss new insights about similarity of offline and online networks.

Further research has been done to identify the interdependence of offline and online information and its possible applications. [?] investigates different methods for the identification of finding real-life contacts based on Twitter data. [START_REF] Subrahmanyam | Online and Offline Social Networks: Use of Social Networking Sites by Emerging Adults[END_REF] investigates social networking of emerging adults' and the relation between online and offline social networks (collected using self-report surveys). [?], [?], [?] explore how the Facebook communications refer to real-life relations, their strength and their changes. Further works consider the task of predicting friendship links in online social networks, e. g., given locations of users from social networks [START_REF] Cranshaw | Bridging the Gap Between Physical Location and Online Social Networks[END_REF], or using time-/geo-tagged photos of the users [START_REF] Crandall | Inferring social ties from geographic coincidences[END_REF]. [START_REF] Eagle | Inferring friendship network structure by using mobile phone data[END_REF] infers friendship network structure (collected using surveys) by using mobile phone data. In contrast to these approaches, this paper focuses on the analysis of structural similarity, inter-relations, dependencies and implications between online and offline social networks. Specifically, we investigate whether the information captured by online social networks can be considered as a valid proxy for offline interactions, i. e., face-to-face proximity networks and scientific collaboration.

III. BACKGROUND

We use graphs for representation, analysis and comparison of social networks. To analyze the content (or text) data we use the bag-of-words model (cf. Section V-B3). We use subgroup discovery to identify and characterize different classes of actors (cf. Section V-C). Below, we summarize the main concepts.

A. Graphs

We represent a network as a graph G = (V, E). A weighted graph is a graph G = (V, E) together with a function w : E ! R + that assigns a positive weight to each edge. For the adjacency matrix

A 2 R n⇥n with n = |V | holds A ij = 1 (A ij = w(i, j)) iff (i, j) 2 E for i, j 2 V ,
= |{j | A ij = 1}| .
In weighted networks, we complement the degree of a node i by its strength s(i) = P j A ij , i. e., the sum of the weights of the attached edges. The Eigenvector centrality (EV-centrality) [?] of a node is an important measure of its influence. The EV-centrality of node depends on the eigenvectors of its neighbors: a node is central, if it has many central neighbors. EV-centrality eig(i) of a node i is defined as follows: eig(i) = P j A ij eig(j), where is a constant. Besides standard statistical measures of correlation and ranking, we apply measures and tests commonly used in social network analysis for comparing graph structures:

• Jaccard Coefficient: For two Graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) with V 1 = V 2 ,
the Jaccard coeffient J of the set of edges E 1 and E 2 is computed as follows:

J(E 1 , E 2 ) = |E1\E2| |E1[E2| .
• QAP test: The quadradic assignment procedure (QAP) test is a standard approach for comparing two graphs G 1 and G 2 ; it is based on estimating the correlation of the respective adjacency matrices [START_REF] Krackhardt | QAP Partialling as a Test of Spuriousness[END_REF]. QAP tests a given graph level statistic, for example the graph covariance, against a QAP null hypothesis. QAP compares the observed graph correlation of (G 1 , G 2 ) to the distribution of the respective resulting correlation scores obtained on repeated random row/column permutations (corresponding to reshuffling of nodes) of the adjacency matrix of G 2 .

B. Bag-of-Words Model

For measuring the semantic similarity of pairs of documents (e. g.,, web pages or scientific papers) -and thus of their authors -in Section V-B3, we make use of the vector space model (bag-of-words-model) of Information Retrieval [START_REF] Manning | Foundations of Statistical Natural Language Processing[END_REF]. Each term of the corpus gives rise to one dimension of a high-dimensional vector space. After applied stemming each document d is represented by a vector ṽ, which at the dimension assigned to term t carries the value tf-idf(d, t)

:= tf(d, t) • log N d(t) with tf(d, t) = f (d,t) max t 0 f (d,t 0 ) , f (d, t
) being the number of occurrences of t in d, N being the total number of documents, and d(t) being the number of documents containing t at least once. The semantic similarity of two documents d 1 and d 2 is then measured by the cosine similarity cosim(d 1 , d 2 ) := cos \( ṽ1 , ṽ2 ) :=

ṽ 1 • ṽ 2 || ṽ 1 ||2•|| ṽ 2 ||2 .

C. Subgroup Discovery

Subgroup discovery, e. g., [START_REF] Atzmueller | Subgroup Discovery[END_REF], [START_REF] Atzmueller | Who are the Spammers? Understandable Local Patterns for Concept Description[END_REF], [START_REF] Wrobel | An Algorithm for Multi-Relational Discovery of Subgroups[END_REF], aims at identifying subgroups of individuals (in our case actors in a social network, cf. Section V-C) that are interesting with respect to a certain target concept. For a binary target concept, for example, we are interested in large subgroups with a high share of individuals for which the target concept is true, e. g., indicating online linking behavior of certain actors (individuals).

A database D = (I, A) is given by a set of individuals I and a set of attributes A. For nominal attributes, a basic pattern

(a i = v j ) is a Boolean function I ! {0, 1} that is true if the value of attribute a i 2 A is equal to v j for the respective individual.
The set of all basic patterns is denoted by ⌃. A subgroup description or (complex) pattern P is given by a set of basic patterns P = {p 1 , . . . , p l }, p i 2 ⌃, i = 1, . . . , l, interpreted as a conjunction p 1 ^. . . ^pl , with length(P ) = l. A pattern can thus also be interpreted as the body of a rule. The rule head then depends on the property of interest, e. g., for a binary target concept T on a basic pattern sel T = true. A subgroup S P := ext(P ) := {i 2 I|P (i) = true}, is the set of all individuals that are covered by the subgroup description P . In a top-k setting, a subgroup discovery algorithm returns the top-k subgroups according to a selectable interestingness measure q : 2 ⌃ ! R , cf. [START_REF] Atzmueller | Subgroup Discovery[END_REF]. For a binary target concept, e. g., the size n := ext(P ) of a subgroup described by the pattern P , i. e., its support, and the share t P of the target concept in the subgroup, i. e., its confidence, are combined by the interestingess measures q S as follows: q S (P ) = n • (t P t 0 ), where t 0 denotes the (default) share of the target concept in the database D, or by the Lift quality function q L (P ) = tP t0 .

IV. DATASETS

We distinguish the data on two dimensions: First, we consider online and offline networks, as well as online and offline content data: We define online data as the data that were not available before the existence of the world wide web (WWW). In this paper, we use two online datasets: the ResearchGate network and homepages of computer scientists. In contrast, offline data represents connections and content whose existence do not specifically depend on the WWW, even if they can be enhanced by WWW-mediated communication now: for that, we consider face-to-face (F2F) interactions during academic conferences and scientific publications. The presented online data was collected from November 2014 till January 2015. The DBLP data was collected in January 2015. Second, we distinguish between network data (faceto-face interactions, follower relations in ResearchGate and scientific publications coauthorship) and content data (content of scientific publications, content of homepages and different attributes in ResearchGate system).

A. Offline: Face-to-Face Proximity

At the LWA 2010, LWA 2011, LWA 2012 conferences (organized by the German Computer Science Society) as well as the ACM Hypertext 2011 conference, we invited participants to wear active RFID devices (proximity tags) that can detect the close-range (about 1.5 meters) face-to-face proximity of the individuals wearing them. This allows us to map out time-resolved networks of face-to-face contacts among the conference attendees.

As in [START_REF] Cattuto | Dynamics of Person-to-Person Interactions from Distributed RFID Sensor Networks[END_REF], we record a face-to-face contact when the length of a contact is at least 20 seconds. A contact ends when the proximity tags do not detect each other for more than 60 seconds. For more information about the proximity sensing technology, we refer the reader to the web site of SocioPatterns 1 . Table I shows high-level statistics of the collected datasets. The duration of each LWA conference was three days, while the Hypertext conference took four days.

B. Offline: Scientific Publications

We collected all data from the DBLP co-authorship network2 in order to determine subgraphs for conference participants. Furthermore, we also retrieved some additional data, e. g., the number of coauthors.

Moreover, we created a corpus of published papers for all 102 persons, who had at least one observed face-to-face (F2F) contact and a ResearchGate profile. We consider all publications listed on a person's DBLP page. For that, we stored paper titles and, when available, also abstracts and fulltexts. In total, we retrieved 4 466 titles, 2 420 abstracts and 2 604 full-texts of 95 of the 102 persons.

In our subsequent analysis using the publication data, we focus on the set of 66 persons, for whom we were able to extract at least one full-text, at least one abstract and at least one title. For these 66 persons, the dataset contains 1 997 fulltexts, 1 858 abstracts and 3 461 titles in total.

C. Online: Homepages

We searched for the homepages of researchers using the Google search engine, using first names and surnames (and the name of the institution if it was necessary) as search terms. The institution name was usually available from the conference proceedings. If researchers had changed their place of work since the conference, we tried to identify both the previous and the current homepages (if available). We considered two languages for the homepages, English and German, since the LWA workshops are visited mainly by German scientists. The Google PageRank was collected for the identified homepages using the Google API. We also used the API of Readability service 3 to extract the page content. This services parses the webpages and uses the modified arc90 algorithm 4 to identify the main content of the page.

D. Online: ResearchGate

Similarly to DBLP, we searched for ResearchGate-profiles of the scientists and used the native built-in ResearchGate search engine for this purpose. We collected the publicly available profile information of the found users (e. g., institution, department), including information that can be considered as user ranking information (e. g., RG-Score, number of profile views etc.), information about their research (disciplines, topics and skills of users) and lists of their followers and followees.

E. Data intersection

Overall more than half of the considered conference participants have a ResearchGate account (102 out of 189) and over two-thirds of them have DBLP entries (146 out of 189) and identified homepages (145 out of 190), cf. Table II. The total number of identified pages is 199 as some researchers have pages in different languages or from different institutions. Almost 40% of all researchers have all three items (ResearchGate account, homepage and DBLP entry).

In order to conduct the experiments for inter-network comparisons, we filtered the data, obtaining a core network representation. Given two networks and their graph respresentations

G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ), the core network for the G 1 network is represented by the graph G ⇤ 1 = (V ⇤ 1 , E ⇤ 1 ), where V ⇤ 1 = {v 2 V 1 |(v, u) 2 E 2 , u 2 V 1 } and E ⇤ 1 = {(v, u)|(v, u) 2 E 1 , v, u 2 V ⇤ 1 }
. Therefore, we removed all nodes that are not active in one of the networks, e. g., some conference participants who have ResearchGate accounts but do not have any connections with other conference participants in this online network are not included in the core network of the conference considering the ResearchGate network.

V. ANALYSIS

In our analysis below, we focus on the three research questions introduced above, i. e., concerning structural similarities and correlation, implications between online and offline, and the characterization of good proxies.

A. Structure and Correlation (Q1)

For answering the first question Q1, we investigate structural similarities of the online and offline data, comparing global network and edge properties. As expected, face-to-face networks are much denser than the other network sources, cf. Tables I). So, we used different contact thresholds (i. e., we kept only links if the duration of at least one face-to-face contact was longer than the considered threshold) to decrease III. The number of active nodes was not reduced significantly in each case.

We compared the Jaccard coefficients of the offline and the online network's cores regarding the edges, cf. Table IV. Moreover, we considered three different minimal contact thresholds for the face-to-face networks. We observe the trend, that the Jaccard similarity is higher for larger minimal contact thresholds, which may point to the fact that stronger ties in offline networks are better mapped in online networks. However, the values of the respective Jaccard coefficients range from 0.15 to 0.32, which does not seem sufficient to state that the two types of networks are very strongly correlated. This is also confirmed by the QAP-test analysis: while almost all correlations are significant, the correlation values themselves are only in a medium range, best for LWA 2010, worst for HT 2011. Furthermore, the Jaccard coefficient between the DBLP coauthorship network and the RG network is in all cases much higher than between the face-to-face proximity and RG networks. This confirms the previous observation that the stronger the connection between two persons, the more likely the presence of the corresponding online connection; obviously, a coauthorship link represents a stronger connection than a conversation during a conference.

The answer to question Q1 can thus be summarized as follows: there exist structural similarities and correlations between offline and online networks. These similarities are much stronger if we consider DBLP instead of the faceto-face networks. Nevertheless, online data tends to act as a better proxy for stronger ties in the face-to-face network compared to all face-to-face interactions. This indicates that strong offline connections are more likely to be represented in online networks than weaker ones. Also, in the offline world, paper coauthorship, for example, is obviously a stronger connection than a conversation during a conference.

B. Implications -Online vs. Offline (Q2)

We investigate implications between the networks, focusing on the recurrance of edges of one network in another network, rankings, and the predictability of offline links given online content.

1) Implication between Sets of Edges:

The implication of edges between different networks helps in further understanding the networks' nature, cf. Table V. The probability that two persons who are linked in ResearchGate also talk to each other during the LWA conference was pretty highabove 80% (69% for the Hypertext 2011 conference). The probability of two persons who are linked in ResearchGate being coauthors is very high as well -over 44%. In contrast, having a short conversation during the conference does not increase the chance for a link in the online world -the probability is below 24%. However, the longer the conversation is, the higher is the probability of the online link -up to 44% (for LWA 2011). The probability of a RG link between two persons in case they coauthored at least one paper is extremely high -86% in average, up to 100% (for LWA 2011). 2) Matching of Rankings: We aim to determine if the hierarchies of different online and offline worlds match each other. To answer this question we identify different features which reflect the "importance" or "influence" of persons in different worlds and compare the resulting ranks afterwards.

As different datasets represent different type of activities and have different nature, we identified different ranking strategies depending on networks properties:

• Face-to-face networks: the participants were ranked according to their eigenvector-centrality (cf. III-A) in each conference (graph); in case the person took part in more than one conference, the maximal EV-centrality was used. • DBLP: the persons with DBLP entries were ranked according to the number of their coauthors; the number of coauthors correlates with the number of publications and is a good sign of collaboration and visibility of a scientist. • ResearchGate: ResearchGate profiles were ranked according to their RG-Score5 -a metric created by the ResearchGate network to compute scientific reputation, based on three parameters: contribution, interaction and reputation.

• Homepages: The homepages of scientists were ranked according to their homepages' Google PageRank; if researchers have different homepages, then the one with the highest PageRank value was chosen.

First, we computed the node rank correlation for all network pairs. The DBLP, ResearchGate and Homepage ranks correlate pairwise and the correlation is significant at p  0.001, cf. Table VI. The largest of all correlations is the correlation between the ResearchGate RG-Score and the DBLP number of coauthors, with a value of 0.68. However, the number of papers and coauthors, respectively, are most probably used in the computation of the RG Score, so these metrics are not independent. Surprisingly, the eigenvector centrality in the face-to-face proximity networks does not correlate with any of these metrics. We considered different centrality measures in the F2F-networks (such as number and lengths of contacts), with similar results (small negative non-significant correlation). Furthermore, we identified the set of the top-ranked persons for each conference using each of the suggested rankings. We considered the top 5 persons obtained from different networks except for the homepage dataset. For that, we adjusted for ties when collecting the top persons according to their homepages' pagerank. Then, we compared the top ranked nodes with each other. Particularly, we computed intersections for each pair of conferences, cf. Table VI. These intersections show how accurately it is possible to predict "important persons" in one network based on data from another network. We computed rankings for the both core and the complete networks. The results are similar to the rank correlation: the intersections between ResearchGate, DBLP and homepages rankings are higher than the intersections of the Face-to-Face proximity networks with each of these networks. However, the top-ranked persons from face-to-face networks are better predictable than we could suggest based on correlations between the rankings, cf. Table VI. Another interesting observation is that comparing the whole networks did not have a crucial influence on the results in most cases. This can be explained by the fact that "important persons" tend to be active in different networks, e. g., having different online and offline manifestations representing their research profiles, interests and publications.

3) Online and Offline Content for Estimating F2F: We analyze the qualities of online and offline data for indicating the existence of face-to-face contacts. This task corresponds to the cold start link prediction problem [START_REF] Leroy | Cold start link prediction[END_REF] for social networks. Given online and offline data, our goal is to retrieve those pairs of conference co-attendees for which a face-to-face contact has been observed at a conference.

We rank all pairs of persons that co-attended a conference by the persons' descending cosine-similarity in five different vectorspaces. We hypothesize, that we find many truly observed F2F contacts early in these rankings. For the first vectorspace, we represent persons by boolean vectors of their selected ResearchGate disciplines. For the second vectorspace, persons are represented by the sums of bag-ofwords vectors built from all their homepages. The other three vectorspaces are based on the data of the persons' publications. We tried bag-of-words vectorspaces from paper full-texts, only abstracts, and only titles. Each vector which represents a person is constructed by summing over the bag-of-words vectors of all the person's publications or homepages. We applied stemming [START_REF] Porter | An algorithm for suffix stripping[END_REF] and stopword removal to publication and homepage data.

We compare the predictive qualities of the rankings from each of these vectorspaces with respect to all four conferences. In particular, we apply microaveraging to the rankings of each vectorspace, i.e., we construct a single multi-conference cosine-ordered ranking for each vectorspace and use this ranking for evaluation. Each triple consists of a conference and two attendees of that conference and is mapped to either "observed" or "not observed" by the respective conference F2F-dataset. We concentrate on those 66 persons, for whom we have data in all of the five vectorspaces, i.e. we know at least one full-text, at least one abstract, at least one title, at least one RG discipline and at least one homepage. For these 66 persons, the dataset contains 1997 full-texts, 1858 abstracts and 3 461 titles in total.

Figure 1a shows the precision and recall curves for retrieving truly observed F2F contacts. The graph indicates that none of the content networks is a good proxy for the face-to-face contact network. Intuitively, this can be explained by many other thinkable reasons to talk to each other aside from similar homepages, publications, or disciplines. Nevertheless, the comparison to the random baseline reveals that utilizing this information is clearly better than not knowing anything. Figure 1a further indicates that retrieving the observed F2F contacts among all pairs of conference co-attenders works comparably well using information from homepages, publication titles or abstracts. At low recall levels, title data yields slightly more precise results. Conversely, ResearchGate disciplines seem to be too broad concepts for achieving high-precision at low recall, but such information works relatively well at higher recall levels.

Because coauthors share publications, their publication based bag-of-words models naturally become more similar. This may also hold for homepages, which frequently contain publication lists. In a second evalutation setup, we examine to what extent the small amount of recall achieved with high precision in Figure 1a can be explained by highly ranked pairs of coauthors. Figure 1b is the result of an evaluation that disregards all coauthor pairs. The graph shows that the Furthermore, homepages are likely to be similar if they belong to the same institution and people from the same institution may be expected to talk to each other with increased probability.

4) Summary: Concluding the analyses of Section V-B, it turned out that there is no definite answer as to whether links and ranks in online networks are good indicator for the existence of the corresponding links and ranks in the offline networks (Q2). Similar to Q1, we observed strong implications between the DBLP and the RG networks in Section V-B1. We also identified highly ranked (important) persons. However, the eigenvector centrality ranking of the face-to-face networks did not show any significant correlation with rankings from the other networks for generalizing these implications.

In Section V-B3, we considered cold start link prediction for the face-to-face networks, an example of a practical application of utilizing online information in order to infer offline relations. We found that information from all the networks can be used to find a small amount of the F2F contacts with high precision and most of the contacts with a clearly better precision than picking links randomly. Interestingly, we found that online data on homepages and in ResearchGate disciplines is better suited for discovering F2F links of non-coauthors than offline paper data.

C. Characteristics of Good Proxies (Q3)

In the following, we analyze features of actors in order to obtain descriptions of certain classes of actors which are good or bad proxies w.r.t. indicating online to offline interactions. We consider both the network and content data separately.

For the network data, we aim to characterize classes of actors for which online connections act as good (or bad) proxies for offline interactions. In particular, we perform the following experiment. We divide all persons who had at least one contact of face-to-face proximity during a given conference and a ResearchGate account into two groups: The first group are given by the persons who have at least one connection in the RG network with another person who visited the conference (the person is a member of core network, cf. Section IV-E). The second group consists of the rest of conference participants -those who do not have connections with other participants in ResearchGate. Thus, we aim to characterize persons whose offline interactions are not discoverable online and vice versa. For that purpose, we applied subgroup discovery [START_REF] Atzmueller | Subgroup Discovery[END_REF], [START_REF] Atzmueller | VIKAMINE -Open-Source Subgroup Discovery, Pattern Mining, and Analytics[END_REF] for identifying subgroups of actors that are characterized by certain descriptive features. We use descriptive information from the presented datasets in order to identify subgroups of users with properties as mentioned earlier, e. g., network centrality measures in the face-to-face contact networks, the Google PageRank of their homepages, and descriptive parameters extracted from their RG profiles. Exemplary results of our analysis are shown in Table VII.

In our analysis, we found interesting properties typical for one or another group. Persons with online connections, for example, have high eigenvector centrality in the face-toface networks (row 1 in the table), high contact fraction with the participants of the conference (row 2), high number of DBLP coauthors (row 3) and high homepage PageRank (row 4). Therefore, the features which associate with importance and connectedness tend to characterize good proxies. This is in line with our conclusions for Q1 and Q2. Another interesting observation is that computer scientists who visited the conferences tend to socialize more online than non-computer scientists who visited these conferences (rows 2, 5, 6, 7). Furthermore, online networks seem to be "bad proxies" for persons with a low eigenvector centrality (rows 6, 10). Also, a low number of RG views as well as a low number of publications (rows 8-9) are indicative.

We also analyzed the matching of the textual content of the homepages of individual scientists with the textual content of their papers. Therefore, we constructed binary variables (as target concepts) indicating a high/low similarity according to the cosine measure of the respective bag-of-word representations. Here, we observe similar important features for good (and bad) proxies as discussed above, e. g., Google PageRank, the number of publications, or the selected ResearchGate disciplines.

Overall, the answer to question Q3 can be summarized as follows: in our analysis, we obtained some interesting properties that can explain the nature of proxies of different quality. We used subgroup discovery to obtain these properties in the form of characteristic descriptions. Alltogether, the presented insights and findings also correspond to the results of the previous experiments.

VI. CONCLUSIONS

In this paper, we analyzed online and offline social networks and their content. We investigated whether information from online sources can be used as a proxy for the offline world, specifically for researchers in the area of computer science. Our results indicate that in many cases online and offline datasets still have structural differences at large. However, strong ties seem to correlate better than weak ones and for "important persons" there are proxy relations between the online and the offline world. We also observed examples of a successful application of online data for offline scenarios, specifically relating to the analysis of content and characteristic subgroups. Our results show that the considered online data is not an ideal proxy for offline information overall, but still provides some important indications about offline relationships.

In the future, we aim to analyze further relations between the presented and additional online (e. g., LinkedIn) and offline (e. g., Meetup.com) data. We also plan to consider timebased phenomena of network development. More and more ubiquitous systems and technologies are emerging, and this makes datasets with more participating persons and their communication in the physical world available. Furthermore, we aim to generalize the presented results (using larger networks and richer content data) for an extended modeling of the correspondence between online and offline social networks.

Fig. 1 :

 1 Fig.1: Precision-recall curves for retrieving existing F2F links from paper, homepage, or RG data. Graph (a) is evaluated on all pairs of conference attendees for which all data is available. Graph (b) further disregards pairs of known coauthors.

TABLE I :

 I High level statistics for the F2F networks: Number of nodes |V | and edges |E|, average degree and strength (weighted degree; an edge e = (u, v) 2 E is weighted by the aggregated duration of all contacts between u, v 2 V ), diameter d, and density.

	Network LWA 10	|V | 77	|E| 1004	?Deg. 26.08	?Str. 20797.22	d 3	Density 0.34
	LWA 11	42	300	14.29	11655.24	3	0.35
	LWA 12	44	354	16.09	14673.32	3	0.37
	HT 11	69	550	15.94	8224.75	4	0.23

TABLE II :

 II Count of items (nodes) in different networks and datasets overall and for particular conferences conferences: F2F -Face-to-face networks; RG -ResearchGate account (RG* -at least one discipline); DBLP -DBLP entry, Ppersons with full-text papers, abstracts, and titles; HP -persons whose homepages were identified (and total number of homepages -some persons have websites in different languages or from different institutions).

		F2F	RG	RG*	DBLP	HP	RG\HP	RG\DBLP	DBLP\HP	RG\DBLP\HP	RG*\HP\P
	LWA 2010	77	50	39	58	56 (79)	42	44	53	41	36
	LWA 2011	42	26	21	37	38 (49)	25	26	35	25	20
	LWA 2012	44	25	23	36	39 (47)	25	24	35	24	22
	HT 2011	69	31	19	55	52 (71)	26	29	45	25	15
	Overall	189	102	74	146	145 (199)	88	93	128	85	66

TABLE III :

 III Networks size and density dependent on conversation threshold for different conferences.

			F2F-Threshold (in sec):
			20:	180:	300:
	LWA 2010	|V | dens(G)	77 0.34	74 0.15	72 0.11
	LWA 2011	|V | dens(G)	42 0.35	39 0.15	39 0.10
	LWA 2012	|V | dens(G)	44 0.37	42 0.17	38 0.14
	HT 2011	|V | dens(G)	69 0.23	61 0.09	58 0.08

TABLE IV

 IV

				F2F -RG
			F2F-Threshold (in sec):	DBLP -RG
			20:	180:	300:
	LWA	Jac	0.14	0.22	0.20	0.79
	2010	QAP	0.29**	0.24**	0.24**	0.75**
	LWA	Jac	0.23	0.31	0.32	0.79
	2011	QAP	0.45**	0.33**	0.29*	0.74**
	LWA	Jac	0.15	0.20	0.26	0.65
	2012	QAP	0.24**	0.21*	0.26**	0.68**
	HT	Jac	0.08	0.16	0.15	0.91
	2011	QAP	0.22**	0.12	0.15	0.64**

: Network correlation analysis: Jaccard coefficient and results of QAP test for pairs of offline (Face-to-face proximity and DBLP-coauthorship) and online networks (ResearchGate) of participants of different conferences. Three thresholds for face-to-face proximity networks were considered: 20, 180 and 300 seconds. Values marked with "**" and "*" are significant at p  0.001 and p  0.05, respectively. the density, as described in [?], cf. Table

TABLE V :

 V Implications: the probability of a link between two nodes in one network if it exists in another for offline (F2F and DBLP coauthorship) and online networks (ResearchGate) of participants of different conferences. We considered three contact thresholds (20, 180 and 300 seconds) for the F2F networks.

				F2F -RG
			F2F-Threshold (in sec):	DBLP -RG
			20:	180:	300:
	LWA		0.84	0.63	0.49	0.46
	2010 LWA	!	0.14 0.81	0.25 0.65	0.26 0.52	0.94 0.54
	2011 LWA	!	0.24 0.83	0.37 0.55	0.44 0.55	1.00 0.44
	2012 HT 2011	! !	0.16 0.69 0.09	0.24 0.53 0.19	0.32 0.38 0.2	0.79 0.44 0.73

TABLE VI :

 VI Correlation of node ranks and intersection of the most important persons (for core (and whole) networks)

	based on different measures: F2F -based on eigenvector
	centrality, DBLP coauthorship -based on number of
	coauthors, ReserchGate network (RG) -based on RG-
	Score, network of webpages -based on global Google
	PageRank value.
			DBLP	RG	HP
	F2F	Correl. Intersec.	0 0.05 (0.05)	-0.07 0.15 (0.05)	-0.01 0.22 (0.22)
	DBLP	Correl. Intersec.		0.68 0.65 (0.6)	0.49 0.4 (0.4)
	RG	Correl. Intersec.			0.42 0.48 (0.32)

TABLE VII :

 VII Examples of descriptions of subgroups of users that connect in Research Gate (RG) online, after at least one offline face-to-face (F2F) contact at a conference, and subgroups of users that do not connect.

	#	Target	Subgroup Description	Support	Confidence	Lift
	1	Connection in RG	Eigenvector centrality (F2F) > 0.28 (high)	0.21	1.00	1.42
	2	Connection in RG	At least one RG CS discipline and contact fraction > 46%	0.17	1.00	1.42
	3	Connection in RG	Number of DBLP coauthors > 67 (high)	0.22	0.96	1.36
	4	Connection in RG	Only RG CS disciplines and Google Pagerank > 3	0.32	0.86	1.22
	5	Connection in RG	At least one RG CS discipline	0.80	0.80	1.14
	6	No RG Connection	No RG CS disciplines and Eigenvector centrality (F2F) < 0.1 (low)	0.14	0.80	2.70
	7	No RG Connection	No RG CS disciplines	0.20	0.68	2.30
	8	No RG Connection	Number of RG views < 547 (low)	0.19	0.62	2.08
	9	No RG Connection	Number of RG publications < 10 (low)	0.22	0.54	1.82
	10	No RG Connection	Eigenvector centrality (F2F) < 0.1 (low)	0.50	0.46	1.56
	accuracy of all models drops when coauthor pairs are not
	evaluated. Yet, the accuracies of the two models from online
	data drop less than those of paper based models. Homepage
	data works best for identifying F2F-contacts of non-coauthors.
	Even ResearchGate disciplines work slightly better than paper
	data on this subtask. An explanation could be that homepages
	and ResearchGate disciplines more concisely describe rough
	topics of research interests and that such interests are important
	features for guessing F2F contacts between non-coauthors.

http://www.sociopatterns.org

http://dblp.uni-trier.de

https://readability.com/developers/api

https://code.google.com/p/arc90labs-readability/

https://www.researchgate.net/publicprofile.RGScoreFAQ.html