Mathieu Moslonka-Lefebvre

Marco Pautasso
email: m.pautasso@ic.ac.uk

Mike J Jeger

Disease spread in small-size directed networks: Epidemic threshold, correlation between links to and from nodes, and clustering

Keywords: Commercial transport, disease management, food webs, initial conditions, spread of ideas, Susceptible-Infected-Susceptible (SIS)

Network epidemiology has mainly focused on large-scale complex networks. It is unclear whether findings of these investigations also apply to networks of small size. This knowledge gap is of relevance for many biological applications, including metacommunities, plant-pollinator interactions and the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. Moreover, many small-size biological networks are inherently asymmetrical and thus cannot be realistically modelled with undirected networks. We modelled disease spread and establishment in directed networks of 100 and 500 nodes at four levels of connectance in six network structures (local, smallworld, random, one-way, uncorrelated and two-way scale-free networks). The model was based on the probability of infection persistence in a node and of infection transmission between connected nodes. Regardless of the size of the network, the epidemic threshold did not depend on the starting node of infection but was negatively related to the correlation coefficient between in-and out-degree for all structures, unless networks were sparsely connected. In this case clustering played a significant role. For small-size scale-free directed networks to have a lower epidemic threshold than other network structures, there needs to be a positive correlation between number of links to and from nodes. When this correlation is negative (one-way scale-free networks), the epidemic threshold for small-size networks can be higher than in non scale-free networks. Clustering does not necessarily have an influence on the epidemic threshold if connectance is kept constant. Analyses of the influence of the clustering on the epidemic threshold in directed networks can also be spurious if they do not consider simultaneously the effect of the correlation coefficient between in-and out-degree.

Introduction

Epidemic models assuming regularly or randomly connected individuals are now involving more complex networks [START_REF] Keeling | The implications of network structure for epidemic dynamics[END_REF][START_REF] May | Network structure and the biology of populations[END_REF][START_REF] Jeger | Modelling disease spread and control in networks: implications for plant sciences[END_REF]. Compared to regular lattices, epidemics in small-world networks are facilitated by long-distance connections [START_REF] Moore | Epidemics and percolation in small-world networks[END_REF]. In scale-free networks of infinite size, epidemics lack a threshold, which implies that even pathogens with a low probability of transmission will persist [START_REF] Pastor-Satorras | Epidemic spreading in scale-free networks[END_REF]. Whether these findings also apply to complex networks of small size is still unclear. This is an important knowledge gap as small-size networks are relevant to many epidemics spreading within subgroups of individuals [START_REF] Liu | The spread of infectious disease on complex networks with householdstructure[END_REF][START_REF] Guimarães | Vulnerability of a killer whale social network to disease outbreaks[END_REF][START_REF] Sun | Dynamical behaviors of epidemics on scale-free networks with community structure[END_REF][START_REF] Pellis | Threshold parameters for a model of epidemic spread among households and workplaces[END_REF].

Networks are not only relevant for epidemiology but have found application in a variety of biological systems [START_REF] Proulx | Network thinking in ecology and evolution[END_REF][START_REF] Bascompte | Networks in ecology[END_REF][START_REF] Jeger | Modelling disease spread and control in networks: implications for plant sciences[END_REF]. Networks with size of the order of magnitude of hundreds of nodes, in turn, are relevant for many current issues in ecology. Examples include closely interacting primate groups (Dunbar, 1993), social networks of manakins [START_REF] Ryder | Social networks in the lek-mating wire-tailed manakin (Pipra filicauda)[END_REF], cavity-nesting community webs [START_REF] Blanc | Cavity-nesting community webs as predictive tools: where do we go from here?[END_REF], diseases of bumble bee colonies [START_REF] Otterstatter | Contact networks and transmission of an intestinal pathogen in bumble bee (Bombus impatiens) colonies[END_REF] and vascular epiphytes on host tree species [START_REF] Lobel | Metapopulation processes in epiphytes inferred from patterns of regional distribution and local abundance in fragmented forest landscapes[END_REF][START_REF] Burns | Network properties of an epiphyte metacommunity[END_REF]. The importance of small-size networks in biology is further shown by plant-pollinator interactions (e.g. [START_REF] Olesen | The smallest of all worlds: pollination networks[END_REF][START_REF] Nielsen | Ecological networks, nestedness and sampling effort[END_REF], mycorrhiza, rhizomorphs and plant pathosystems [START_REF] Southworth | Application of network theory to potential mycorrhizal networks[END_REF][START_REF] Lamour | Network formation by rhizomorphs of Armillaria lutea in natural soil: their description and ecological significance[END_REF][START_REF] Brooks | Spatial and temporal heterogeneity explain disease dynamics in a spatially explicit network model[END_REF], and food webs (e.g. [START_REF] Montoya | Small world patterns in food webs[END_REF][START_REF] Neutel | Reconciling complexity with stability in naturally assembling food webs[END_REF].

Although networks of small size are relevant both for epidemiology and for ecology and evolution (e.g. [START_REF] Hanski | The metapopulation capacity of a fragmented landscape[END_REF][START_REF] Brooks | Quantifying population substructure: extending the graph-theoretic approach[END_REF][START_REF] Ings | Ecological networks -beyond food webs[END_REF], there has been surprisingly little theoretical work investigating whether results obtained for large-size and infinite networks also apply to small-size networks. We know for example that heterogeneity in the contact structure can markedly lower invasion thresholds in networks of large size [START_REF] Boguna | Cut-offs and finite size effects in scale-free networks[END_REF][START_REF] Colizza | Invasion threshold in heterogeneous metapopulation networks[END_REF][START_REF] Jeger | Modelling disease spread and control in networks: implications for plant sciences[END_REF], but there is still only patchy knowledge about the dynamics and properties of small-size networks of various structures. This is of concern, given the several applications of small-size networks in natural systems and also given that the rapid globalization and structural changes of trade interactions are increasing the relevance of complex networks of small size for the invasion biology of exotic organisms [START_REF] Jones | Introductions of non-native plant pathogens into Great Britain, 1970-2004[END_REF][START_REF] Dehnen-Schmutz | A century of the ornamental plant trade and its impact on invasion success[END_REF][START_REF] Brenn | Community structure of Phialocephala fortinii s. lat. in European tree nurseries, and assessment of the potential of the seedlings as dissemination vehicles[END_REF]. A real world application of directed networks of small size in epidemic controlling are trade movements of infected material amongst plant nurseries (e.g. [START_REF] Bandyopadhyay | Contemporary global movement of emerging plant diseases[END_REF][START_REF] Anderson | Emerging infectious diseases of plants: pathogen pollution, climate change and agrotechnology drivers[END_REF][START_REF] Slippers | Emerging pathogens: fungal host jumps following anthropogenic introduction[END_REF]. A recent example is given by the spread in regional networks of plant nurseries and garden centres of Phytophthora ramorum, the oomycete causing Sudden Oak Death in the West

Coast of the USA and leaf blight and dieback in many ornamental shrubs both in America and Europe [START_REF] Werres | Phytophthora ramorum sp nov., a new pathogen on Rhododendron and Viburnum[END_REF][START_REF] Prospero | Population dynamics of the sudden oak death pathogen Phytophthora ramorum in Oregon from 2001 to 2004[END_REF][START_REF] Xu | Spatio-temporal analysis of Phytophthora ramorum cases in England and Wales[END_REF]. Given the wide range of species affected, movements of infected plant material in the horticultural trade have the potential to make this emerging plant disease outbreak even more widespread [START_REF] Holdenrieder | Tree diseases and landscape processes: the challenge of landscape pathology[END_REF][START_REF] Frankel | Sudden oak death and Phytophthora ramorum in the USA: a management challenge[END_REF][START_REF] Grünwald | Phytophthora ramorum: a pathogen with a remarkably wide host range causing sudden oak death on oaks and ramorum blight on woody ornamentals[END_REF]. There is thus the necessity to control the network of plant nurseries and retail outlets trading ornamental species susceptible to P. ramorum in an effective and efficient way.

In this study, we investigated whether heterogeneity in the contact structure and the presence of short-cuts as in small-world networks still make a difference to epidemic development in small-size, directed networks. Directed networks, given the more complicated adjacency matrices, have been used to model epidemics relatively rarely [START_REF] Newman | Random graphs with arbitrary degree distributions and their applications[END_REF][START_REF] Meyers | Predicting epidemics on directed contact networks[END_REF][START_REF] Park | Dynamic behaviors in directed networks[END_REF][START_REF] Kenah | Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing[END_REF], but are relevant to many real-world situations with asymmetries in contact structures, and deserve more study for various network sizes, structures and levels of connectance. We have shown elsewhere (Pautasso and Jeger, 2008) that heterogeneity in the contact structure still affects the epidemic threshold even in the case of networks of one hundred nodes, but inquire here whether variations in the epidemic threshold (the boundary between no epidemic and an epidemic) for different network structures and at different levels of connectance can be explained by the correlation coefficient between links to and from nodes and by the clustering coefficient of the network. Previous work in percolation theory suggests that the correlation coefficient between links to and from nodes is important in the case of heterogeneity in the contact structure [START_REF] Schwartz | Percolation in directed scale-free networks[END_REF];

Materials and Methods

We simulated disease spread and establishment in networks of 100 and 500 nodes. For both network sizes, we used six kinds of structure: (1) local (nearest-neighbour transmission), ( 2) random (nodes connected with probability p), (3) small-world (local networks rewired with short-cuts), and scale-free structure (see [START_REF] Jeger | Modelling disease spread and control in networks: implications for plant sciences[END_REF] for a visualization). For scale-free networks, we considered separately networks with in-and out-degree of nodes (4) positively, ( 5) not and (6) negatively correlated. The networks were directed, i.e. a link from node a to node b did not imply the reverse connection (as e.g. in [START_REF] Newman | Random graphs with arbitrary degree distributions and their applications[END_REF][START_REF] Boguna | Generalized percolation in random directed networks[END_REF][START_REF] Meyers | Predicting epidemics on directed contact networks[END_REF][START_REF] Park | Dynamic behaviors in directed networks[END_REF]. Directed networks are realistic approximations of many real world systems; wherever a directed link does not entail the reverse connection (asymmetrical interactions), from food webs to plant-animal mutualistic networks, from infectious disease epidemiology to the spread of information [START_REF] Bascompte | The nested assembly of plant-animal mutualistic networks[END_REF][START_REF] Da Gama | Epidemics in small world networks[END_REF][START_REF] Bode | Using complex network metrics to predict the persistence of metapopulations with asymmetric connectivity patterns[END_REF][START_REF] Thebault | Does asymmetric specialization differ between mutualistic and trophic networks?[END_REF].

For each network structure, 100 replicates were built in MATLAB at each level of connectance (for 100 nodes: 100, 200, 400, and 1000 links; for 500 nodes: 500, 2500, 5000, and 10000 links; Table 1). Rather than keeping connectance constant between the two network sizes, we needed a lower level of connectance for 500 nodes (500 links, i.e. connectance = 0.002) to obtain results similar to the lower level of connectance for 100 nodes (100 links, i.e. connectance = 0.01). To obtain a disappearance of a significant correlation between epidemic threshold and correlation links in and out for the larger network size (500 nodes), it was necessary to use a lower connectance level that at network size equal 100 nodes. We also left out results from the highest level of connectance for 500 nodes (25000 links, i.e. connectance = 0.1) as in this case networks were overly connected so that differences between network structures were likely to be small. Local networks were built starting from a regular ring with 100 (for 100 nodes) and 500 (for 500 nodes) links more than the This rewiring coefficient allowed the construction of small-world networks with clustering intermediate between those of random and of local networks. Small-world networks are networks with high clustering compared to random networks and small shortest path length compared to local networks, and these conditions were respected in our case. Scalefree networks were built with a preferential attachment algorithm, starting with a seed network and based on five parameters adding nodes and/or links depending on the in-, out-, and total degree of existing nodes:

1) The parameter a added both a node i and an arrow (i,j) according to the in-degree of an existing node j. This step was repeated a-times.

2) The parameter b added both a node i and an arrow (j,i) according to the out-degree of an existing node j. This step was repeated b-times.

3) The parameter c added both a node i and an arrow (i,j) or (j,i) according to the total degree of an existing node j. The choice of the direction of each arrow added, i.e. (i,j) or (j,i), was equally probable. This step was repeated c-times.

4) The parameter d only added an arrow (i,j) and not a node according to the outdegree of an existing node i and to the in-degree of an existing node j (ij). This step was repeated d-times.

5)

The parameter e only added an arrow (i,j) and not a node according to the total degrees of existing nodes i and j (ij). This step was repeated e-times.

Epidemic development was deterministic, with discrete time-step and governed by the probabilities of infection transmission between nodes (p t ) and of infection persistence in a node (p p ). The transmission probability p t was either zero (unconnected nodes) or a value constant for different links in a network but variable amongst network replicates (in order to work at the threshold conditions in each replicate; see Pautasso and Jeger, 2008). The persistence probability p p combined in one single parameter the length of infectiousness, detection and control measures. We also set p p to be the same for all nodes. Both p t and p p are real variables, going from 0 to 1. This can be a realistic assumption for many ecological networks, wherever persistence and transmission are not either switched on or off, but can assume any value between these two extremes. We assumed all nodes to be of equal capacity and kind (differences between nodes are thus entirely due to their in-and outdegree). At each iteration, the contact structure of the network realization was maintained exactly the same. Networks were not necessarily fully connected, so it is possible that at the lower levels of connectance not all nodes could be reached from all nodes.

For each iteration, we obtained the infection status of a given node P i (x) in the following way:

P i (x) = Σ p t (x,y) P i (y)
for y going from 1 to 100, where p t refers to the connection of the node x from a node y, and

P i (y)
is the infection status of the node y at the previous iteration. At the beginning of the epidemic P i (x) was set to zero for all nodes except for the starting node of the epidemic, with P (i) = 1. For the connection of a node with itself, p p was used instead of p t . The biological motivation for self-loops is that nodes which have become infected by a pathogen have a certain probability to remain infected due to the persistence of inoculum through time. The model was thus a SIS (Susceptible-Infected-Susceptible) model. This can be a realistic assumption for many epidemiological systems, wherever nodes are still at risk even after eradication of a disease outbreak if complete immunization is not possible and if there is a continuing trade or contact with susceptible material or inoculum [START_REF] Jeger | Modelling disease spread and control in networks: implications for plant sciences[END_REF].

The development of the epidemic was assessed on the basis of the sum of P i (x) across all nodes and on the basis of the number of nodes with P i (x) higher than an arbitrary value (0.01). The epidemic was started with a single infection of a single node, as the threshold conditions were not affected by whether epidemics are started with a single or with multiple infections (unpublished observations). Also, results were consistent using a different starting probability of infection. Although the starting node had a marked influence on the epidemic size at equilibrium (Pautasso and Jeger, 2008), making the epidemic start from different nodes did not affect the threshold conditions (p* p and p* t ) which define a boundary between no epidemic and an epidemic. Given that there is a linear threshold in a graph of p* p as a function of p* t (Pautasso and Jeger, 2008), we worked at p* p = 0 and assessed the threshold only in terms of p* t .

The clustering coefficient C i of a node i which is part of a digraph characterizes the extent to which nodes adjacent to any node i are adjacent to each other. More precisely,

where Card (X) symbolyses the cardinality of the set X, i.e. the number of elements of X.

The neighbourhood ƥ i of a vertex i is the digraph that consists solely of the set V ƥi of vertices connected from and/or to i (not including i itself) and of the set E ƥi of all arrows connecting such vertices. The average clustering C of a digraph is the average of the clustering of each node of this digraph.

Analysis of variance (ANOVA) of the epidemic threshold and of the correlation coefficient between in-and out-degree of the 100 nodes of the network replicates for the different levels of connectance (within a network structure) and for the different types of network structure (at a given level of connectance) was carried out in SAS 9.1 (proc ANOVA). The same package (proc GLM) was used for multi-variate regressions of the threshold p* t against the correlation coefficient between in-and out-degree and the average clustering coefficient of the 100/500 nodes of the network replicate for a given network structure and level of connectance.

Results

The threshold p* t significantly decreased with increasing connectance for all structures and with both network sizes (Fig. 1a,b). With the exception of the lowest connectance level for both network sizes, two-way scale-free networks showed a significantly lower and oneway scale-free networks a significantly higher threshold than all other structures (Fig. 1c, threshold not significantly different than the one of local networks (Fig. 1d). Small-world networks showed a threshold not significantly different from random networks (except at the lowest connectance for both network sizes). The threshold of uncorrelated scale-free networks, at all connectance levels and for both network sizes, lay between those for twoway and one-way scale-free networks (Fig. 1c,d).

The correlation coefficient between in-and out-degree of the nodes of the networks was, by definition, positive for two-way scale-free networks, not significantly different from zero for uncorrelated scale-free networks (except, by chance, at some levels of connectance; Fig. 1e,f), and negative for one-way scale-free networks. This coefficient was also clustered around zero for local, small-world and random networks. There was a significant increase in the correlation coefficient between in-and out-degree for two-way scale-free networks with increasing connectance for both network sizes (Fig. 1g,h).

With the exception of the lowest connectance, there were generally significantly negative relationships between threshold p* t and the correlation coefficient between in-and outdegree for all structures and with both network sizes (Fig. 2; Tables 2,3). At the lowest level of connectance, these relationships were significant only for small-world and twoway scale-free networks for the network size of 100 nodes. The proportion of variance in threshold p* t explained by the correlation between the in-and the out-degree increased with connectance for all structures and for both network sizes, but with exception of twoway scale-free networks. Apart from one-way scale-free networks for 500 nodes and twoway scale-free networks for both network sizes, at the highest connectance these proportions of variance were substantial (between 0.84 and 0.93 for 100 nodes, and between 0.56 and 0.95 for 500 nodes). However, apart from one-way scale-free networks at the network size of 100 nodes, the slopes of these relationships tended to become flatter with increasing connectance (Tables 2,3). This is a consequence of the overall lower threshold p* t at higher connectance (Fig. 2). Including the clustering as an additional explaining factor in the regression of the threshold p* t against the correlation coefficient between links to and from nodes did not affect the previous results. The clustering increased with increasing level of connectance for all structures and for both network sizes (Figs. 3,4). By definition, at a given connectance, the clustering was higher in local than in small-world networks, and in small-world compared to random networks (Figs. 3,4). The clustering also decreased from two-way to uncorrelated and to one-way scale-free networks (Figs. 3,4). For both network sizes, the clustering was significantly negatively related to the threshold p t at the lowest level of connectance for local and small-world networks, and at all levels of connectance for two way scale-free networks (Fig. 3; Tables 2,3).

For both network sizes, the relationship of the correlation coefficient between links to and from nodes with the clustering was significantly positive for all scale-free networks (except at the lowest level of connectance; Fig. 4; Tables 4,5). However, the proportion of variance in the correlation coefficient between in-and out-degree explained by the clustering was substantial for uncorrelated and one-way, but not for two-way scale-free networks. For both network sizes, there was also a significantly positive correlation between these two variables for random networks at the two highest connectance levels (Tables 4,5).

Discussion

Networks of small size have biological significance in a variety of ecological fields.

Examples include meta-populations, mutualistic and antagonistic interactions [START_REF] Dunne | Food-web structure and network theory: the role of connectance and size[END_REF][START_REF] Lundgren | The dense and highly connected world of Greenland's plants and their pollinators[END_REF][START_REF] Brooks | Quantifying population substructure: extending the graph-theoretic approach[END_REF]Pautasso et al., 2008;[START_REF] Thebault | Does asymmetric specialization differ between mutualistic and trophic networks?[END_REF]. In spite of the relevance of small-size networks for many issues in natural sciences, it is not clear whether theoretical results derived from analyses of large-scale complex networks apply also to small-size networks [START_REF] Guimarães | Vulnerability of a killer whale social network to disease outbreaks[END_REF]. Moreover, much work in network epidemiology has focused on undirected networks (e.g. [START_REF] Keeling | The implications of network structure for epidemic dynamics[END_REF][START_REF] Shirley | The impact of network topology on disease spread[END_REF][START_REF] May | Network structure and the biology of populations[END_REF], whereas many of the biological networks mentioned in the Introduction are inherently asymmetrical and thus can only be realistically modelled with directed networks. Our analysis shows that in SIS models of epidemics in directed networks of small size, the threshold is lower for scale-free network structures only if there is a positive correlation between in-and out-degree of nodes (Fig. 2). This finding is in good agreement with results obtained in percolation theory for large-size networks [START_REF] Schwartz | Percolation in directed scale-free networks[END_REF]. Our work further shows that when this correlation between in-and out-degree of nodes is negative (one-way scale-free networks), for small-size networks the epidemic threshold is higher than in non scale-free networks. This result is broadly independent of the connectance level and of the network size, although it breaks down in case of sparsely connected networks.

For sparsely connected networks, differences in clustering amongst networks can become important.

Clustering has been shown to be a network feature relevant to the development and control of epidemics (e.g. [START_REF] Eames | Contact tracing and disease control[END_REF][START_REF] Keeling | The implications of network structure for epidemic dynamics[END_REF][START_REF] Kiss | Disease contact tracing in random and clustered networks[END_REF][START_REF] Hartivgsen | Network structure, and vaccination strategy and effort interact to affect the dynamics of influenza epidemics[END_REF][START_REF] Turner | A network model of E. coli O157 transmission within a typical UK dairy herd: the effect of heterogeneity and clustering on the prevalence of infection[END_REF][START_REF] Miller | Spread of infectious disease through clustered populations[END_REF]. More clustered networks are believed to be less prone to invasion by a pathogen, as this will be likely to become confined inside clusters of connected nodes [START_REF] Keeling | The implications of network structure for epidemic dynamics[END_REF][START_REF] Naug | Structure of the social network and its influence on transmission dynamics in a honeybee colony[END_REF]. Clusters can also slow down epidemic development in the first phases of epidemics (e.g. [START_REF] Szendroi | Polynomial epidemics and clustering in contact networks[END_REF]. In scale-free networks of infinite size, the presence of high local clustering has been shown to make it possible for a non-null epidemic threshold to be present [START_REF] Eguiluz | Epidemic threshold in structured scale-free networks[END_REF]; but see [START_REF] Serrano | Percolation and epidemic thresholds in clustered networks[END_REF]. Clustering in large-scale scale-free networks has been shown to decrease the size of epidemics, but also to decrease their threshold [START_REF] Newman | Properties of highly clustered networks[END_REF].

In the case of our small-size networks, the influence of clustering on the epidemic threshold is dwarfed by the one of the correlation coefficient between in-and out-degree.

The negative correlation between epidemic threshold and clustering for uncorrelated and one-way scale-free networks at the highest levels of connectance (Fig. 3) can be explained by the positive correlation of the correlation coefficient between in-and out-degree with the clustering in the same networks (Fig. 4). Unlike [START_REF] Eames | Modelling disease spread through random and regular contacts in clustered populations[END_REF] for undirected networks, we do not observe any differences in the influence of clustering on epidemic development depending on whether contacts are regular or random. In our small-size, directed networks, with the exception of sparsely connected local networks, clustering has no influence on the epidemic threshold within both local and random networks (Fig. 3).

Our main conclusion is that, in directed networks, analyses of the influence of clustering on the epidemic threshold can be spurious if they do not consider simultaneously the effect of the correlation coefficient between in-and out-degree. In some cases (sparsely connected local, small-world and two-way scale free networks), we observe a negative correlation between epidemic threshold and clustering and no significant effect of the correlation coefficient between in-and out-degree on the threshold, suggesting that in extreme situations the correlation coefficient between in-and out-degree may not play the same role as for large-scale networks. We also point out that analyses which show that clustering is negatively related to the epidemic threshold (e.g. [START_REF] Britton | Epidemics on random graphs with tunable clustering[END_REF] need to take the connectance level into account, as (i) the clustering increases with increasing connectance, (ii) the epidemic threshold decreases with increasing connectance, so that (iii) if connectance increases the epidemic threshold is bound to be negatively correlated with the clustering. Our results show, however, that, when keeping connectance constant, clustering and epidemic threshold are not necessarily related.

Epidemics in scale-free networks are now commonly investigated (e.g. [START_REF] Boccaletti | Complex networks: structure and dynamics[END_REF][START_REF] Masuda | Multi-state epidemic processes on complex networks[END_REF][START_REF] Colizza | Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations[END_REF], but there has been less attention to directed scale-free networks. In this analysis, we confirm the importance of the correlation coefficient between in-and out-degree for epidemics in directed scale-free networks [START_REF] Woolhouse | Epidemiological implications of the contact network structure for cattle farms and the 20-80 rule[END_REF]. We also provide evidence for a different behaviour of the clustering in directed scale-free networks where there are different correlation coefficients between in-and out-degree. Although there is a positive correlation of the clustering with the correlation coefficient between in-and out-degree for all scale-free network types (except at the lowest connectance level), only for uncorrelated and one-way scale-free networks has this correlation a substantial r 2 (for 500 nodes: between 0.54 and 0.79, Table 5). For two-way scale-free networks, the correlation between clustering and correlation coefficient between links in and out of nodes is significant but negligible (again for 500 nodes, r 2 values are between 0.01 and 0.08, Table 5). Further work is needed to investigate whether results obtained for undirected scale-free networks apply to all types of directed scale-free networks.

Our results are essentially independent of the network size used (100 and 500 nodes). We have avoided using a lower number of nodes than 100 as it is likely that differences between network structures will tend to become blurred for even smaller networks.

However, an interesting question would be at which small size heterogeneity in the contact structure stops having a significant influence on network properties and epidemic processes. A remarkable result is also that network breakdown at the lowest connectance level (which causes for example the disappearance of a significant relationship between epidemic threshold and correlation coefficient between in-and out-degree) happens at two different connectance levels (0.01 for 100 nodes and 0.002 for 500 nodes, Table 1) for the two network sizes. More research is needed to assess which other processes are not independent of the interaction between network size and connectance.

In conclusion, our study confirms the result obtained for large-size networks that regardless of the size of the network, the epidemic threshold is negatively related to the correlation coefficient between in-and out-degree for all structures. However, this does not hold if small-size networks are sparsely connected. In this case, clustering plays a significant role (Table 2). In small-size networks, when the correlation between links in and out is negative (one-way scale-free networks), the epidemic threshold can be higher than in non scale-free networks. Contrary to what is found in analyses of large-size networks, clustering does not necessarily have an influence on the epidemic threshold of small-size networks if connectance is kept constant. Analyses of the influence of the clustering on the epidemic threshold in directed networks can also be spurious if the effect of the correlation coefficient between in-and out-degree is disregarded.

  -size networks with different (i) size, (ii) structure (local, random, small-world and scale-free), (iii) levels of connectance, and (iv) clustering.

  links and by randomly generating 100 or 500 gaps, respectively. Random digraphs were generated using the G(N,M) model where M directed links are placed randomly and independently between the N nodes of the graph. Small-world networks were built with the Watts and Strogatz (1998) algorithm and a rewiring coefficient of 0.25.

  network size of 100 nodes, random networks showed a significantly lower threshold than local networks, but not at the highest connectance (Fig1c). This result was not confirmed at the larger network size, where random networks had an epidemic

Fig. 1 :

 1 Fig. 1: Threshold probability of transmission for the different (a, b) levels of connectance and (c, d) network structures, and correlation coefficient between in-and out-degree of the nodes of the 100 replicates for the different (e, f) levels of connectance and (g, h) network structures. Network size = 100 (a, c, e, g) and 500 (b, d, e, h) nodes. Error bars are standard deviations of 100 replicates. Different letters show significant differences (ANOVA, p < 0.05) within (a, b, e, f) levels of connectance for a given network structure, and (c, d, g, h) network structures for a given level of connectance.

Fig. 2 :

 2 Fig. 2: Threshold probability of transmission for the different network replicates as a function of the correlation coefficient between in-and out-degree of the nodes for the six network structures. Network size = 100 nodes (number of links = (a) 100, (b) 200, (c) 400, and (d) 1000) and 500 nodes (number of links = (e) 500, (f) 2500, (g) 5000, and (h) 10000).

Fig. 3 :

 3 Fig. 3: Threshold probability of transmission for the different network replicates as a function of the average clustering coefficient of the nodes for the six network structures. Network size = 100 nodes (number of links = (a) 100, (b) 200, (c) 400, and (d) 1000) and 500 nodes (number of links = (e) 500, (f) 2500, (g) 5000, and (h) 10000).

Fig. 4 :

 4 Fig. 4: Correlation coefficient between in-and out-degree for the different network replicates as a function of the average clustering coefficient of the 100 nodes for the six network structures. Network size = 100 nodes (number of links = (a) 100, (b) 200, (c) 400, and (d) 1000) and 500 nodes (number of links = (e) 500, (f) 2500, (g) 5000, and (h) 10000).

Table 1 .

 1 Number of links (L) and connectance (C = L/N 2 ) for the two network sizes studied (number of nodes = N).

	N	100
	L	100	200	400	1000
	C	0.01	0.02	0.04	0.1
	N	500
	L	500	2500	5000	10000
	C	0.002	0.01	0.02	0.04

Table 2 .

 2 Proportion of variance explained, slope and associated p value of the regression of threshold probability of transmission as a function of (a) the correlation coefficient between in-and out-degree and (b) the clustering for the 100 replicates of the six network structures (local, random, small-world, two-way, uncorrelated, and one-way scale-free) at four levels

	of connectance (100, 200, 400 and 1000 links; network size = 100 nodes).
			100			200			400			1000
		r 2	a	b	r 2	a	b	r 2	a	b	r 2	a	b
	l	0.17	+0.05	-5.45	0.33	-0.06	+0.09	0.58 -0.01	+0.03	0.85	-0.001	+0.000
		p	0.30	0.001		0.001	0.15		0.001	0.03		0.001	0.98
	r	0.02	-0.07	-0.76	0.54	-0.25	+0.04	0.74 -0.06	+0.02	0.86	-0.009	-0.003
		p	0.23	0.41		0.001	0.89		0.001	0.79		0.001	0.86
	sw 0.22	-0.16	-4.63	0.27	-0.11	-0.02	0.62 -0.02	-0.00	0.84	-0.002	+0.001
		p	0.003	0.001		0.001	0.86		0.001	0.82		0.001	0.45
	sf2 0.11	-0.17	-3.24	0.55	-0.23	-0.30	0.34 -0.09	-0.02	0.25	-0.04	-0.01
		p	0.008	0.04		0.001	0.001		0.001	0.05		0.001	0.001
	sf0 0.01	-0.04	-0.90	0.28	-0.59	-0.27	0.59 -0.69	+0.28	0.93	-0.27	+0.02
		p	0.49	0.48		0.001	0.61		0.001	0.33		0.001	0.45
	sf1 0.04	+0.12	-1.52	0.03	-0.86	+1.66	0.23 -3.31	+4.09	0.87	-0.92	+0.09
		p	0.27	0.15		0.12	0.24		0.001	0.005		0.001	0.35

Table 3 .

 3 Proportion of variance explained, slope and associated p value of the regression of threshold probability of transmission as a function of (a) the correlation coefficient between in-and out-degree and (b) the clustering for the 100 replicates of the six network structures (local, random, small-world, two-way, uncorrelated, and one-way scale-free) at four levels

	of connectance (500, 2500, 5000 and 10,000 links; network size = 500 nodes).
			500			2500			5000			10000
		r 2	a	b	r 2	a	b	r 2	a	b	r 2	a	b
	l	0.15	+0.00	-32.7	0.66	-0.01	-0.00	0.48 -0.001 -0.001 0.56	-0.0001 -0.0001
		p	0.97	0.001		0.001	0.63		0.001	0.68		0.001	0.25
	r	0.02	-0.21	+3.43 0.77	-0.04	-0.02	0.87 -0.01	-0.03	0.94	-0.002	-0.003
		p	0.21	0.50		0.001	0.89		0.001	0.31		0.001	0.66
	sw 0.24	-0.13	-26.7	0.66	-0.01	+0.00	0.68 -0.002 -0.003 0.87	-0.0004 -0.0001
		p	0.20	0.001		0.001	0.85		0.001	0.15		0.001	0.76
	sf2 0.08	-0.20	-18.1	0.50	-0.06	-0.09	0.60 -0.03	-0.02	0.38	-0.04	-0.004
		p	0.06	0.03		0.001	0.001		0.02	0.001		0.001	0.001
	sf0 0.01	-0.12	-6.20	0.41	-3.66	+5.74	0.83 -0.64	+0.44	0.95	-0.16	+0.07
		p	0.48	0.44		0.001	0.01		0.001	0.001		0.001	0.001
	sf1 0.01	-0.12	-5.11	0.15	-4.16	+2.59	0.55 -2.28	+0.75	0.40	-0.63	+0.11
		p	0.38	0.49		0.001	0.33		0.001	0.07		0.001	0.07

Table 4 .

 4 Proportion of variance explained, slope and associated p value of the regression of the correlation coefficient between in-and out-degree as a function of the clustering for the 100 replicates of the six network structures (local, random, small-world, two-way, uncorrelated, and one-way scale-free) at four levels of connectance (100, 200, 400 and 1000 links; network size = 100 nodes).

			100			200			400			1000
		r 2	slope	p	r 2	slope	p	r 2	slope	p	r 2	slope	p
	l	0.06	-7.40	0.01	0.00	-0.01	0.99	0.00 +0.26	0.85	0.00	-0.93	0.88
	r	0.00	-0.65	0.69	0.01	+0.14	0.37	0.07 +5.37	0.008	0.06	+10.1	0.01
	sw 0.00	-0.84	0.68	0.01	-0.49	0.41	0.03 +1.55	0.09	0.00	+0.33	0.83
	sf2 0.00	+1.46	0.56	0.18	+1.13	0.001	0.04 +0.16	0.04	0.06	+0.05	0.01
	sf0 0.00	-0.03	0.98	0.13	+1.81	0.001	0.65 +2.61	0.001	0.70	+1.83	0.001
	sf1 0.05	-2.02	0.03	0.08	+0.71	0.005	0.42 +1.50	0.001	0.69	+1.37	0.001

Table 5 .

 5 Proportion of variance explained, slope and associated p value of the regression of the correlation coefficient between in-and out-degree as a function of the clustering for the 100 replicates of the six network structures (local, random, small-world, two-way, uncorrelated, and one-way scale-free) at four levels of connectance (500, 2500, 5000 and 10000 links; network size = 500 nodes).

			500			2500			5000			10000
		r 2	slope	p	r 2	slope	p	r 2	slope	p	r 2	slope	p
	l	0.05	-45.5	0.03	0.00	+1.61	0.48	0.00 -3.40	0.46	0.00	-4.90	0.52
	r	0.01	+3.08	0.32	0.02	+7.42	0.21	0.09 +22.4	0.002	0.06	+25.0	0.01
	sw 0.00	+2.67	0.59	0.00	+0.05	0.96	0.00 +0.02	0.98	0.00	-0.84	0.60
	sf2 0.00	-0.37	0.96	0.08	+0.19	0.005	0.09 +0.04	0.002	0.01	+0.01	0.33
	sf0 0.00	+2.82	0.55	0.73	+3.33	0.001	0.79 +2.18	0.001	0.71	+1.71	0.001
	sf1 0.00	-1.63	0.75	0.54	+1.60	0.001	0.74 +1.10	0.001	0.55	+0.45	0.001

Acknowledgments

Many thanks to T. Harwood, O. Holdenrieder, J. Parke, M. Shaw, J. Tufto, F. Van den Bosch and X. Xu for discussions and insights, and to T. Matoni and anonymous reviewers for helpful comments on a previous version of the ms. This study was funded by the Department for Environment, Food and Rural Affairs, the Rural Economy and Land Use Programme, UK, and the French Ministry of National Education and Research.


Vincent Andrieu

Christophe Prieur

Sophie Tarbouriech

Denis Arzelier

Synthesis of a global asymptotic stabilizing feedback law for a system satisfying two different sector conditions

Global asymptotic stabilization for a class of nonlinear systems is addressed. The dynamics of these systems are composed of a linear part to which is added some nonlinearities which satisfy two different sector bound conditions depending wether the state is closed or distant from the origin. The approach described here is based on the uniting of control Lyapunov functions as introduced in [2]. The stabilization problem may be recast as an LMI optimization problem for which powerful semidefinite programming softwares exist. This is illustrated by a numerical example.

I. INTRODUCTION

There is an extensive literature on the design of nonlinear stabilizers providing numerous techniques which apply on specific classes of nonlinear systems. The class of system under interest in this paper is the one described by nonlinear functions satisfying sector bound conditions. This class of nonlinearity includes many different memoryless functions (see e.g., [START_REF] Khalil | Nonlinear Systems[END_REF]Chapter 6] for an introduction on this topic) such as saturations (see e.g. [START_REF]Advanced strategies in control systems with input and output constraints[END_REF], [START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions[END_REF], [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF] for design techniques of control system with such nonlinearities). To oppose to what has been done in these papers, two different sector conditions are considered: one sector condition when the state is near the equilibrium and one other sector condition when the state is distant from the equilibrium. This distinction between small and large values of the distance from the state to the equilibrium allows us to better describe the nonlinear system. Moreover we remark that encompassing both sector conditions into one global sector condition may lead to a too conservative synthesis problem which may not have a solution (see the example of Section V-A below).

This motivates us to consider the local sector condition and the non-local one separately and to design successively 1) a local stabilizer with a basin of attraction containing a compact set and 2) a non-local controller such that the previous compact set is globally attractive. After that, to design a global stabilizer, we suggest to piece together both controllers. Many different techniques exist to unit two different feedback laws. Let us cite the use of hybrid controllers to unit them (see [START_REF] Prieur | Uniting local and global controllers with robustness to vanishing noise[END_REF], [START_REF] Prieur | Uniting a high performance, local controller with a global controller: the output feedback case for linear systems with input saturation[END_REF], [START_REF] Prieur | Quasi-optimal robust stabilization of control systems[END_REF], [START_REF] Teel | Uniting local and global controllers[END_REF]). In the present paper we apply the technique introduced in [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] where a continuous solution to the uniting problem is given through the construction of a uniting control Lyapunov function 1 .

vincent.andrieu@gmail.com, cprieur@laas.fr, tarbour@laas.fr, arzelier@laas.fr 1 see [START_REF] Arststein | Stabilization with relaxed controls[END_REF] for a definition of CLF.

In [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] some conditions are given to provide a global stabilizer from a local control Lyapunov function and a nonlocal one 2 . In the following, we succeed to rewrite these sufficient conditions in terms of LMIs and we introduce a global stabilizer for the control of systems satisfying two different sector conditions.

To be more precise, consider the system defined by its state-space equation:

ẋ = 𝐴 𝑥 + 𝐵 𝑢 + 𝐺 𝜙(𝐿 𝑥) (1)
where the state vector 𝑥 is in ℝ 𝑛 . (𝐴, 𝐵, 𝐺, 𝐿) are matrices respectively in ℝ 𝑛×𝑛 , ℝ 𝑛×𝑚 , ℝ 𝑛×𝑝 and ℝ 𝑞×𝑛 . Moreover 𝑢 in ℝ 𝑚 is the control input and 𝜙(𝑧) : ℝ 𝑞 → ℝ 𝑝 is a nonlinear locally Lispchitz function such that 𝜙(0) = 0.

One way to design a global stabilizer for system (1) is to use circle and Popov criteria (see [START_REF] Arcak | Circle and Popov criteria as tools for nonlinear feedback design[END_REF]) under the assumption that the nonlinear function 𝜙 satisfies some sector bound conditions 3 :

(𝜙(𝐿𝑥) -𝑀 𝐿𝑥) ′ (𝜙(𝐿𝑥) -𝑁 𝐿𝑥) ≤ 0 , ∀ 𝑥 ∈ ℝ 𝑛 , (2)
where 𝑀 and 𝑁 are two given matrices in ℝ 𝑞×𝑝 . Following4 [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF], a constructive LMI condition allowing to design a state feedback control law solving the stabilizing problem may be exhibited.

The aim of this paper is to study the case in which the function 𝜙 satisfies two different sector conditions depending on the distance between 𝑥 and the origin. The idea of the design is then to apply techniques inspired by [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] to unite local and non-local controllers and to provide a global stabilizer.

Assumptions on the nonlinear function 𝜙 introduced in (1) can be given as follows:

Assumption 1: Local sector condition. There exist a positive real number 𝑣 0 , two matrices (𝑀 0 , 𝑁 0 ) in ℝ 𝑝×𝑞 × ℝ 𝑝×𝑞 such that, for all |𝑥| ≤ 𝑣 0 , we have:

(𝜙(𝐿 𝑥) -𝑀 0 𝐿 𝑥) ′ (𝜙(𝐿 𝑥) -𝑁 0 𝐿 𝑥) ≤ 0 . ( 3
)
2 see [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] for definitions of local and non-local CLFs. 3 Note that other classes of sector conditions are possible. In particular we may consider the generalized sector conditions written as

(𝜙(𝑧) -𝑀 𝑧) ′ 𝐷 (𝜙(𝑧) -𝑁 𝑧) ≤ 0 ,
where 𝐷 is any given diagonal positive definite matrix (as in [START_REF] Gomes Da | Antiwindup design with guaranteed regions of stability : an LMI-based approach[END_REF], [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF]). Despite the fact that considering such generalized sector conditions is possible, we restrict our attention to sector bound condition as [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] to ease the exposition of our results.

Assumption 2: Non-local sector condition. There exist a positive real number 𝑣 ∞ < 𝑣 0 , two matrices (𝑀 ∞ , 𝑁 ∞ ) in ℝ 𝑝×𝑞 × ℝ 𝑝×𝑞 such that, for all |𝑥| ≥ 𝑣 ∞ , we have:

(𝜙(𝐿 𝑥) -𝑀 ∞ 𝐿 𝑥) ′ (𝜙(𝐿 𝑥) -𝑁 ∞ 𝐿 𝑥) ≤ 0 . ( 4
)
System satisfying both Assumptions 1 and 2 is of interest since local and non-local approximations of nonlinear global dynamics may be found in the literature (see for instance [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF]). Moreover, as shown in the example introduced in Section V, it might be useful to split a global sector condition in two pieces (a local and a non-local one) in order to get a solution where the usual LMI-based sufficient conditions obtained from [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF] are too conservative.

In this paper, we address the following problem: Problem: Under Assumptions 1 and 2, can we design a nonlinear control law 𝑢 = 𝛼(𝑥) with 𝛼 : ℝ 𝑛 → ℝ 𝑚 a continuous function ensuring global asymptotic stabilization of the origin for the system (1) ?

Before considering this global stabilization problem, we first consider each sector separately in Section II and we design a local (resp. a non-local) controller 𝑢 = 𝛼 0 (𝑥) (resp. 𝑢 = 𝛼 ∞ (𝑥)) using the local (resp. non-local) sector condition of Assumption 1 (resp. of Assumption 2). After this step, a new controller, which is equal to the local controller 𝑢 = 𝛼 0 (𝑥) on a neighborhood of the origin and equal to the non-local controller 𝑢 = 𝛼 ∞ (𝑥) outside a compact set, is designed. This construction is based on [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF] and is considered in Section III. We then formalize a sufficient condition, expressed in terms of the existence of solutions to LMIs constraints, allowing us to address the global stabilization problem in one step in Section IV. Two numerical examples illustrate the previous results in Section V, and Section VI contains some concluding remarks.

Note that all proofs have been removed due to space limitation and can be obtained from the authors upon request.

Notation. The Euclidian norm is denoted by | ⋅ |. For a positive real number 𝑛, 𝐼 𝑛 (resp. 0 𝑛,𝑚 ) denotes the identity matrix (resp. the null matrix) in ℝ 𝑛×𝑛 (resp. in ℝ 𝑛×𝑚 ). The subscripts may be omitted when there is no ambiguity. Moreover for a vector 𝑥 the diagonal matrix defined by the entries of 𝑥 is noted Diag(𝑥), and for a matrix 𝑀 , Sym(𝑀 ) = 𝑀 + 𝑀 ′ .

II. DESIGN OF LOCAL AND OF NON-LOCAL

CONTROLLERS

A. Local case

In this section, we consider Assumption 1 and we design a state feedback ensuring local asymptotic stabilization of the origin for the system [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF]. Note that if we introduce:

𝐴 0 = 𝐴 + 𝐺𝑀 0 𝐿 , 𝜙 0 (𝑧) = 𝜙(𝑧) -𝑀 0 𝑧 , 𝑆 0 = 𝑁 0 -𝑀 0 ,
system (1) can be rewritten as:

ẋ = 𝐴 0 𝑥 + 𝐵 𝑢 + 𝐺 𝜙 0 (𝐿 𝑥) , (5)
and the local sector condition becomes

𝜙 0 (𝐿𝑥) ′ (𝜙 0 (𝐿𝑥) -𝑆 0 𝐿𝑥) ≤ 0 , ∀ |𝑥| ≤ 𝑣 0 . ( 6
)
Hence, inspired by [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF], we can state a sufficient condition to get local asymptotic stabilization of the origin: Proposition 2.1: Suppose Assumption 1 is satisfied (hence (6) holds). If there exists a symmetric positive definite matrix 𝑊 0 in ℝ 𝑛×𝑛 , two matrices 𝐻 0 in ℝ 𝑚×𝑛 , and 𝐽 0 in ℝ 𝑚×𝑝 satisfying the LMI:

[ Sym(𝐴 0 𝑊 0 + 𝐵𝐻 0 ) ★ 𝐽 ′ 0 𝐵 ′ + 𝐺 ′ + 𝑆 0 𝐿𝑊 0 -2𝐼 𝑝 ] < 0 , ( 7
)
then the control law 𝑢 = 𝛼 0 (𝑥) where

𝛼 0 (𝑥) = 𝐾 0 𝑥 + 𝐽 0 𝜙 0 (𝐿 𝑥) , ( 8
)
with 𝐾 0 = 𝐻 0 𝑊 -1 0 makes the origin of the system a locally asymptotically stable equilibrium, with basin of attraction containing the set

ℰ(𝑊 -1 0 , 𝑅 0 ) = {𝑥 ∈ ℝ 𝑛 , 𝑥 ′ 𝑊 -1 0 𝑥 ≤ 𝑅 0 }
where 𝑅 0 is any positive real number satisfying

𝑅 0 𝑊 0 -𝑣 2 0 𝐼 𝑛 ≤ 0 . ( 9
)

B. Non-local case

A result similar to Proposition 2.1 can be obtained when considering Assumption 2. Indeed, if we introduce:

𝐴 ∞ = 𝐴 + 𝐺𝑀 ∞ 𝐿 , 𝜙 ∞ (𝑧) = 𝜙(𝑧) -𝑀 ∞ 𝑧 , 𝑆 ∞ = 𝑁 ∞ -𝑀 ∞ ,
system (1) becomes:

ẋ = 𝐴 ∞ 𝑥 + 𝐵 𝑢 + 𝐺 𝜙 ∞ (𝐿 𝑥) , ( 10
)
and the non-local sector condition (i.e. inequality (4)) becomes:

𝜙 ∞ (𝐿𝑥) ′ (𝜙 ∞ (𝐿𝑥) -𝑆 ∞ 𝐿𝑥) ≤ 0 , ∀ |𝑥| ≥ 𝑣 ∞ . ( 11
)
We can state a sufficient condition to get global asymptotic stabilization of a set containing the origin: Proposition 2.2: Suppose Assumption 2 is satisfied (hence [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF]

holds). If there exists a symmetric positive definite matrix 𝑊

∞ in ℝ 𝑛×𝑛 , two matrices 𝐻 ∞ in ℝ 𝑚×𝑛 , and 𝐽 ∞ in ℝ 𝑚×𝑝 satisfying the LMI: [ Sym(𝐴 ∞ 𝑊 ∞ + 𝐵𝐻 ∞ ) ★ 𝐽 ′ ∞ 𝐵 ′ + 𝐺 ′ + 𝑆 ∞ 𝐿𝑊 ∞ -2𝐼 𝑝 ] < 0 , ( 12
)
then the control law 𝑢 = 𝛼 ∞ (𝑥) where

𝛼 ∞ (𝑥) = 𝐾 ∞ 𝑥 + 𝐽 ∞ 𝜙 ∞ (𝐿 𝑥) , ( 13
)
with 𝐾 ∞ = 𝐻 ∞ 𝑊 -1
∞ makes the solutions of the closed-loop system complete and the set

ℰ(𝑊 -1 ∞ , 𝑟 ∞ ) = {𝑥 ∈ ℝ 𝑛 , 𝑥 ′ 𝑊 -1 ∞ 𝑥 ≤ 𝑟 ∞ } ,
globally and asymptotically stable where 𝑟 ∞ is any positive real number such that In this section, we assume that we have solved the local stabilization problem and the non-local one following Propositions 2.1 and 2.2. Hence, we have in hand the matrices 𝑃 0 , 𝑃 ∞ , 𝐾 0 , 𝐾 ∞ , 𝐽 0 and 𝐽 ∞ such that the LMI ( 7) and ( 12) are satisfied (with 𝑃 0 = 𝑊 -1 0 and 𝑃 ∞ = 𝑊 -1 ∞ ) and both controllers 𝛼 0 and 𝛼 ∞ defined by ( 8) and ( 13) respectively. We wish now to unite these two controllers to get a controller making the origin a global and asymptotic stable equilibrium.

𝑣 2 ∞ 𝐼 𝑛 -𝑊 ∞ 𝑟 ∞ ≤ 0 . ( 14
)
To solve this problem, we follow the uniting strategy introduced in [START_REF] Andrieu | Uniting two control Lyapunov functions for affine Systems[END_REF]. Following this procedure, the first step is to unite the local CLF 𝑥 → 𝑥 ′ 𝑃 0 𝑥 and the non-local one 𝑥 → 𝑥 ′ 𝑃 ∞ 𝑥. In order to do this, we need an extra Assumption expressing the fact that the two sets, in which we have a stability property, overlap.

Assumption 3: Covering Assumption. There exist two positive real numbers 𝑅 0 and 𝑟 ∞ such that ( 9) and ( 14) are satisfied and such that

𝑟 ∞ 𝑃 0 -𝑅 0 𝑃 ∞ < 0 . ( 15
)
In Figure 1, an illustration of the covering Assumption is presented (it is used the numerical values of Section V-A). This assumption implies that we have the following inclusions:

{𝑥, |𝑥| ≤ 𝑣 ∞ } ⊆ ℰ(𝑃 ∞ , 𝑟 ∞ ) ⊂ ℰ(𝑃 0 , 𝑅 0 ) ⊆ {𝑥, |𝑥| ≤ 𝑣 0 } .
To get a global stabilizing control law, we have the following result:

Theorem 3.1: Assume that Assumptions 1, 2 and 3 hold. If there exist two matrices 𝐾 𝑚 in ℝ 𝑚×𝑛 and 𝐽 𝑚 in ℝ 𝑚×𝑝 , and four positive real numbers (𝜇 1 , 𝜇 2 , 𝜇 3 , 𝜇 4 ) such that the following LMIs are satisfied:

[ Sym(𝑃 0 [𝐴 + 𝐵𝐾 𝑚 ]) ★ (𝐽 ′ 𝑚 𝐵 ′ + 𝐺 ′ )𝑃 0 0 ] -𝜇 1 𝑄 0 -𝜇 2 𝑄 ∞ < 0 , ( 16
) [ Sym(𝑃 ∞ [𝐴 + 𝐵𝐾 𝑚 ]) ★ (𝐽 ′ 𝑚 𝐵 ′ + 𝐺 ′ )𝑃 ∞ 0 ] -𝜇 3 𝑄 0 -𝜇 4 𝑄 ∞ < 0 , ( 17
)
where

𝑄 0 = [ Sym(𝐿 ′ 𝑀 ′ 0 𝑁 0 𝐿) ★ -(𝑀 0 + 𝑁 0 ) 𝐿 2𝐼 𝑝 ] , 𝑄 ∞ = [ Sym(𝐿 ′ 𝑀 ′ ∞ 𝑁 ∞ 𝐿) ★ -(𝑀 ∞ + 𝑁 ∞ ) 𝐿 2𝐼 𝑝 ] , ( 18
)
then there exists a continuous function 𝛼 : ℝ 𝑛 → ℝ 𝑚 such that the control law 𝑢 = 𝛼(𝑥) makes the origin a globally asymptotically stable equilibrium for system [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF].

A possible control law ensuring a global asymptotic stabilization of the origin of system (1) is given in [2, Proposition 4] and is expressed as:

𝛼(𝑥) = ℋ(𝑥) -𝑘 𝑐(𝑥) ∂𝑉 ∂𝑥 (𝑥)𝐵 (19)
with ℋ a continuous function such that

ℋ(𝑥) = { 𝛼 0 (𝑥) if 𝑉 ∞ (𝑥) ≤ 𝑟 ∞ , 𝛼 ∞ (𝑥) if 𝑉 0 (𝑥) ≥ 𝑅 0 .
The function 𝑐 is any continuous function such that5

𝑐(𝑥)

{ = 0 if 𝑉 0 (𝑥) ≥ 𝑅 0 or 𝑉 ∞ (𝑥) ≤ 𝑟 ∞ , > 0 if 𝑉 0 (𝑥) < 𝑅 0 and 𝑉 ∞ (𝑥) > 𝑟 ∞ .
𝑘 is a positive real number sufficiently large and finally 𝑉 is a global CLF (CLF) for system (1) obtained following the procedure introduced in [2, Theorem 1] which enables to unite both CLFs 𝑉 0 and 𝑉 ∞ .

IV. DESIGN IN ONE STEP

Following the design strategy exposed in the previous sections, a stabilizing control law for system (1) satisfying Assumptions 1 and 2 can be performed if we succeed in solving the following algorithm:

Design separately a local and a non-local CLF (i.e. 𝑃

0 = 𝑊 -1 0 and 𝑃 ∞ = 𝑊 -1
∞ ) via the LMIs ( 7) and ( 12), and check if 1) they satisfy the covering Assumption [START_REF] Skelton | A unified approach to linear control design[END_REF] with 𝑅 0 and 𝑟 ∞ satisfying ( 9) and ( 14); 2) they satisfy the LMIs feasibility conditions [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] and [START_REF]Advanced strategies in control systems with input and output constraints[END_REF] to be united.

In this section, we investigate the possibility of solving this problem in one shot. In other words, we wish to find an LMI formulation to prove the existence of matrices 𝑃 0 and 𝑃 ∞ satisfying the conditions in items 1) and 2) of the previous algorithm.

⎡ ⎣ Sym(𝐴𝑆 + 𝐵𝐻 𝑚 ) ★ ★ 𝑆 -𝑊 0 + 𝐴𝑆 + 𝐵𝐻 𝑚 -2𝑊 0 𝐵𝐽 𝑚 + 𝐺 -𝑊 0 𝑀 0 𝐿𝑆 + 𝑀 ∞ 𝐿𝑆 + 𝐽 ′ 𝑚 𝐵 ′ + 𝐺 ′ + 𝐴𝑆 + 𝐵𝐻 𝑚 ★ -4𝐼 𝑛 -Sym(𝐵𝐽 𝑚 + 𝐺) ⎤ ⎦ < 0 (20) ⎡ ⎣ Sym(𝐴𝑆 + 𝐵𝐻 𝑚 ) ★ ★ 𝑆 -𝑊 ∞ + 𝐴𝑆 + 𝐵𝐻 𝑚 -2𝑊 ∞ 𝐵𝐽 𝑚 + 𝐺 -𝑊 ∞ 𝑀 0 𝐿𝑆 + 𝑀 ∞ 𝐿𝑆 + 𝐽 ′ 𝑚 𝐵 ′ + 𝐺 ′ + 𝐴𝑆 + 𝐵𝐻 𝑚 ★ -4𝐼 𝑛 -Sym(𝐵𝐽 𝑚 + 𝐺) ⎤ ⎦ < 0 (21)
A. About the covering Assumption Assumption 3 may fail when considering an arbitrary pair of matrices 𝑃 0 and 𝑃 ∞ computed using Propositions 2.1 and 2.2.

Moreover, note that inequality [START_REF] Skelton | A unified approach to linear control design[END_REF], combined with Propositions 2.1 and 2.2, is not linear in 𝑃 0 or 𝑃 ∞ since 𝑅 0 and 𝑟 ∞ depend on 𝑃 0 and 𝑃 ∞ through the constraints ( 9) and [START_REF] Prieur | Quasi-optimal robust stabilization of control systems[END_REF]

in which 𝑊 0 = 𝑃 -1 0 and 𝑊 ∞ = 𝑃 -1
∞ . Nevertheless, note that, when 𝑅 0 = 𝑟 ∞ , the covering Assumption can be easily defined as the following LMI:

𝑊 0 -𝑊 ∞ > 0 . ( 22
)
Moreover, note that the two matrix inequality constraints ( 9) and ( 14) can be recast as the following LMI constraints:

𝜌𝑣 2 ∞ 𝐼 𝑛 -𝑊 ∞ ≤ 0 , 𝑊 0 -𝜌𝑣 2 0 𝐼 𝑛 ≤ 0 ( 23
)
where 𝜌 is a positive real number such that 𝑅 0 = 𝑟 ∞ = 1 𝜌 . Consequently, this feasibility constraint can be added easily in the design of 𝑊 0 and 𝑊 ∞ (i.e. of 𝑃 0 and 𝑃 ∞ ).

To summarize, we have: Proposition 4.1: Suppose there exist two positive definite matrices 𝑊 0 , 𝑊 ∞ in ℝ 𝑛×𝑛 and a real number 𝜌 > 0 such that inequalities ( 22) and ( 23) are satisfied, then the covering Assumption (i.e. inequality [START_REF] Skelton | A unified approach to linear control design[END_REF]) is also satisfied with 𝑅 0 = 𝑟 ∞ = 1 𝜌 .

B. About the second feasibility constraint

Now to include the feasibility constraints ( 16) and ( 17) into the design of a global asymptotic stabilizer, we need to restrict ourselves to a specific class of matrices 𝑊 0 , 𝐻 0 , 𝐽 0 , 𝑊 ∞ , 𝐻 ∞ , and 𝐽 ∞ solutions of ( 7), ( 12), ( 22), (23).

To be more precise we consider the subclass of solutions such that the conditions ( 16) and ( 17) are satisfied, by particularizing these conditions as LMI conditions. To do that we use elimination lemma [START_REF] Skelton | A unified approach to linear control design[END_REF], and we get the following result.

Proposition 4.2: If the local and the non-local conditions (3) and ( 4) are such that 𝑁 0 = 𝑁 ∞ = 0 and if there exist two symmetric positive definite matrices 𝑊 0 and 𝑊 ∞ in ℝ 𝑛×𝑛 , two matrices 𝐽 𝑚 in ℝ 𝑚×𝑝 and 𝐻 𝑚 in ℝ 𝑚×𝑛 , an invertible matrix 𝑆 in ℝ 𝑛×𝑛 , satisfying the LMI constraints ( 20) and ( 21), then inequalities ( 16) and [START_REF]Advanced strategies in control systems with input and output constraints[END_REF]

hold with 𝑃 0 = 𝑊 -1 0 , 𝑃 ∞ = 𝑊 -1 ∞ , 𝐾 𝑚 = 𝐻 𝑚 𝑆 -1 and 𝜇 1 = 𝜇 2 = 𝜇 3 = 𝜇 4 = 1.
Note that employing a change of matrix similar to the one introduced in Section II-A and II-B, we can also deal with the case where 𝑁 0 = 𝑁 ∞ ∕ = 0.

The key point of the previous result is that the constraints (20) and ( 21) obtained are linear in the unknown 𝑊 0 and 𝑊 ∞ . Consequently with Propositions 4.1 we are able to give a complete LMI formulation allowing us to design a state feedback control law for system (1) making the origin a globally and asymptotically stable equilibrium. This can be summarized as follows.

Theorem 4.3: If 𝑁 0 = 𝑁 ∞ = 0 and if there exist two symmetric positive definite matrices 𝑊 0 and 𝑊 ∞ in ℝ 𝑛×𝑛 , three matrices 𝐽 0 , 𝐽 ∞ and 𝐽 𝑚 in ℝ 𝑚×𝑝 three matrices 𝐻 0 , 𝐻 ∞ and 𝐻 𝑚 in ℝ 𝑚×𝑛 , an invertible matrix 𝑆 in ℝ 𝑛×𝑛 , and a real number 𝜌 > 0 satisfying the matrix inequality ( 7), ( 12), ( 20), ( 21), ( 22), and ( 23), then the control law 𝑢 = 𝛼(𝑥) with 𝛼 defined in [START_REF] Teel | Uniting local and global controllers[END_REF] makes the origin a globally asymptotically stable equilibrium for system [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF].

V. TWO EXAMPLES

To illustrate the proposed procedure and to motivate it, we present two examples. The first one exhibits the interest of splitting in two pieces a global sector bound condition, the other one is an illustration of Theorem 4.3.

A. Example 1

In a first example, we consider system (1) described by the following data:

ẋ = 𝜙(𝑥) + 𝐵 𝑢 , 𝑥 ∈ ℝ 3 , (24)
with

𝐵 = ⎡ ⎣ 1 0 0 ⎤ ⎦ .
To complete the definition of system (1), it remains to introduce the function 𝜙 : ℝ 3 → ℝ 3 representing the nonlinearity of the system. First consider both matrices 𝑀 ∞ in ℝ 3×3 and 𝑁 0 in ℝ 3×3 defined as

𝑀 ∞ = ⎡ ⎣ 0 0 0 0 0 1 1 0 0 ⎤ ⎦ , 𝑁 0 = ⎡ ⎣ 0 0 0 1 0 0 0 1 0 ⎤ ⎦ .
The nonlinear function 𝜙 is defined as a locally Lipschitz continuous path interpolating 𝑀 ∞ and 𝑁 06 , i.e:

𝜙(𝑥) = 𝜆(|𝑥|)𝑀 ∞ 𝑥 + (1 -𝜆(|𝑥|))𝑁 0 𝑥 , 𝑥 ∈ ℝ 3 , ( 25
)
where 𝜆 : ℝ + → ℝ + is an increasing locally Lipschitz function such that 7 :

𝜆(0) = 0 , lim 𝑠→+∞ 𝜆(𝑠) = 1 .
We wish to find a controller guaranteeing global asymptotic stabilization of the origin.

1) Employing a unique and global sector condition:

A first strategy to address the stabilization problem for system (24) is to check if the solvability conditions inspired by [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF] are satisfied. For this purpose, let us first prove that the nonlinear function 𝜙 satisfies a (global) sector condition: 8 Proposition 5.1: The function 𝜙 defined in (25) satisfies the sector condition:

(𝜙(𝑥) -𝑀 ∞ 𝑥) ′ (𝜙(𝑥) -𝑁 0 𝑥) ≤ 0 , ∀ 𝑥 ∈ ℝ 3 .
So following the computations of (the proof of) Proposition 2.1 (or applying [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF] without saturation or [START_REF] Montagner | Robust absolute stability and nonlinear state feedback stabilization based on polynomial luré functions[END_REF]), we may try to compute a global nonlinear feedback law by solving the following LMI optimization problem: Proposition 5.2: If there exists a symmetric positive definite matrices 𝑊 in ℝ 3×3 , and two matrices 𝐻 in ℝ 1×3 , 𝐽 in ℝ satisfying the LMI:

[ Sym(𝑀 ∞ 𝑊 + 𝐵𝐻) ★ 𝐽 ′ 𝐵 ′ + 𝐼 3 + (𝑁 0 -𝑀 ∞ )𝑊 -2𝐼 1 ] < 0 ,
then the control law:

𝑢(𝑥) = 𝐾 𝑥 + 𝐽 𝜙(𝑥) , ( 26
)
with 𝐾 = 𝐻𝑊 -1 makes the origin a globally and asymptotically stable equilibrium for System (24). Using parser YALMIP [START_REF] Lofberg | YALMIP: Yet Another LMI Parser[END_REF] and LMI solver SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], this problem is found to be unfeasible on this particular instance. This approach is clearly too conservative and cannot be used in this specific case.

2) Employing the uniting controller approach: Note however that our uniting controller provides another approach to solve this stabilizing problem. First we show that the function 𝜙 introduced in (25) fits in the context of Assumptions 1 and 2.

Proposition 5.3: Given two positive real numbers 0 < 𝜆 ∞ < 𝜆 0 < 1, Assumptions 1 and 2 are satisfied with

𝑣 0 = 𝜆 -1 (𝜆 0 ) , 𝑣 ∞ = 𝜆 -1 (𝜆 ∞ ) , 𝑀 0 = 𝜆 0 𝑀 ∞ + (1 -𝜆 0 )𝑁 0 , and 𝑁 ∞ = 𝜆 ∞ 𝑀 ∞ + (1 -𝜆 ∞ )𝑁 0 .
Consequently, we are in the framework of the uniting sector condition.

Note that for this system we have 𝑁 0 ∕ = 0 and 𝑁 ∞ ∕ = 0 (and 𝑁 0 ∕ = 𝑁 ∞ ), and thus Theorem 4.3 cannot be applied. Therefore we follow the procedure developed in Sections III and IV-A, and we apply Propositions 2.1, 2.2, and 4.1. Therefore we have to check if the LMIs (7), ( 12), ( 22) and (23) have a solution in 𝑊 0 , 𝑊 ∞ , and 𝜌 (among others variables), and we get that Assumption 3 holds for 𝑃 0 = 𝑊 -1 0 , 𝑃 ∞ = 𝑊 -1 ∞ , and 𝑅 0 = 𝑟 ∞ = 1 𝜌 . Choosing 𝜆 0 = 0.6 and 𝜆 ∞ = 0.4, and assuming that 𝜆 is a continuous function such that:

𝑣 0 = 𝜆 -1 (0.6) = 10 , 𝑣 ∞ = 𝜆 -1 (0.4) = 1.
⎤ ⎦ 𝜌 -1 = 𝑅 0 = 𝑟 ∞ = 16.3666
of the LMIs ( 7), ( 12), ( 22) and ( 23). The fact that the covering Assumption is satisfied for the matrix 𝑃 0 and 𝑃 ∞ with the positive real number 𝑣 0 = 10 and 𝑣 ∞ = 1.5 is guaranteed by Proposition 4.1 and is illustrated in Figure 1.

Hence we are in the context of Theorem 3.1 and we can check that, for the two previous matrices 𝑃 0 and 𝑃 ∞ , there exist two matrices 𝐽 𝑚 , 𝐾 𝑚 and four scalars 𝜇 𝑖 satisfying the sufficient conditions ( 16) and ( 17). This is indeed the case with 𝐽 𝑚 = [-1.0000 -5.5737 -5.3139] ,

and

𝐾 𝑚 = [1.0491 -0.7931 -0.6987].
Consequently, the conclusion of Theorem 3.1 holds, we get that the control law ( 19) makes the origin a globally and asymptotic equilibrium for the system (24).

B. Example 2

We consider now a second order system defined as:

{ ẋ1 = 𝑥 2 + 𝑥 1 𝜂 1 (𝑥) , ẋ2 = 𝑢 + 𝑥 2 𝜂 2 (𝑥) , ( 27
)
where the functions 𝜂 1 and 𝜂 2 are any continuous functions such that:

𝜂 1 (𝑥) { ≤ 1 if |𝑥| ≤ 𝑣 0 ≤ 2 if |𝑥| ≥ 𝑣 0 , 𝜂 2 (𝑥) { ≤ 1 if |𝑥| ≤ 𝑣 ∞ ≤ 0.5 if |𝑥| ≥ 𝑣 ∞ .
Note that this system can be written in the form (1) with:

𝐴 = [ 0 1 0 0 ] , 𝐵 = [ 0 1
] ,

with 𝐺 = 𝐿 = 𝐼 2 , and with the function

𝜙(𝑥) = [ 𝑥 1 𝜂 1 (𝑥) 𝑥 2 𝜂 2 (𝑥)
] .

The function 𝜙(𝑥) satisfies Assumptions 1 and 2 with 𝑁 0 = 𝑁 ∞ = 0 , 𝑀 0 = 𝐼 2 , 𝑀 ∞ = Diag(2, 0.5) .

Hence we are in the context of Theorem 4.3. Employing YALMIP [START_REF] Lofberg | YALMIP: Yet Another LMI Parser[END_REF] with the solver SEDUMI [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], we check the solvability of LMIs ( 7), ( 12), ( 22), ( 23), ( 20) and (21). These LMIs are solvable with: and finally 𝜌 = 0.0681. Consequently, the conclusion of Theorem 4.3 holds and we obtain a control law making the origin a globally and asymptotic stable equilibrium for the system (27).

𝑊 0 = [ 0,

VI. CONCLUSION

In this paper we have introduced the synthesis problem of a nonlinear feedback law for a class of control systems. The control systems under consideration are those with a nonlinearity satisfying a sector condition when the state is close to the equilibrium and a (maybe) different sector condition when the state is distant from the equilibrium. We noted that encompassing both sector conditions into a unique global one may lead to a too conservative synthesis problem. This motivates us to consider both properties of the nonlinearities separately and to design successively 1) a local asymptotic stabilizing nonlinear controller whose basin of attraction contains some compact set and 2) a non-local controller which makes the previous compact set globally attractive. Then we compute a nonlinear controller which piece together the local controller with the non-local one, and we obtain a global asymptotic stabilizing controller. We emphasize that the sufficient conditions to solve this design problem are written in terms of LMIs. Two numerical examples motivate and illustrate this approach.

Fig. 1 .

 1 Fig. 1. Illustration of the Covering Assumption

  5 , 8 Note that the proof of Proposition 5.1 does not use the expression of the system (1) but only properties of the nonlinearity 𝜙 as introduced in (25).

	we get the following solutions
	𝑃 0 =	⎡ ⎣	0.3635 0.7820 0.6798 0.7820 3.4710 2.7149	⎤ ⎦
			0.6798 2.7149 2.6948
	𝑃 ∞ =	⎡ ⎣	0.4085 0.8650 0.8252 0.8650 3.7691 3.0466
			0.8252 3.0466 3.7480

All authors are with LAAS-CNRS, Université de Toulouse, 7, avenue du Colonel Roche, 31077 Toulouse, France. Vincent Andrieu is also with LAGEP, CNRS, CPE, Université Lyon 1, Villeurbanne, France.

Note that in[START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF] is addressed a more involved problem since some saturations on the input are considered.

For instance, with Assumption 3 we can take:𝑐(𝑥) = max{0, (𝑅 0 -𝑉 0 (𝑥))(𝑉∞(𝑥) -𝑟∞)} .

This is typically the case when a system is modeled as an interpolation of several linear systems.

For instance 𝜆 can be defined by 𝜆(𝑠) = 2 𝜋 arctan(𝑠) for all 𝑠 ≥ 0.