Available online at www.sciencedirect.com

SciVerse ScienceDirect PrOC6d ¡0

Social and Behavioral Sciences

Procedia - Social and Behavioral Sciences 79 (2013) 82- 100

9th Conference on Applications of Social Network Analysis (ASNA)

Detecting Communities within French Intercorporate Network

Sana Elouaer-Mrizaka, Marc Chastandb

aUniversity of Jean Monnet Saint Etienne, GATE-LSE, 6 Rue Basse des Rives, 42100 Saint Etienne France & Faculty of Economics Sousse (Lamided), Tunisia b Magellan, IAE of Lyon III, 69007 Lyon, France

Abstract

People sitting on the same board have the possibility to meet and interact. Therefore, directors holding many directorships constitute a link between boards. Overlap in group membership allows for the flow of information between groups, and perhaps coordination of the group's actions. In comparison to the body of literature on interlocking directorates, there are relatively few studies on the network of French interlocks. In our study, and to detect community structure within the French board network among the companies listed in the French Financial Index CAC40, we use the maximum modularity approach. To our knowledge, this is the only study on French interlocks that employees a maximum modularity approach. The main result is that the network shows a strong community structure.

This is due to several reasons. On the one hand, we have the existence directors serving in several boards. Generally, this means that the "small world" of French listed companies is actually split into identifiable communities. On the other hand, we have motives to control and reduce environmental uncertainty. Overlap in group membership allows for the flow of information between groups, and perhaps coordination of the group's actions. According to Sonquist and Koenig (1975), interlocking among boards through common directors make easily coordination among firms. Lang and Lockhart (1990) justified that a firm creates a connection through an interlock to guarantee access to external resource. Our results show that main financial institutions are present among the large extracted communities.

© 2013 The Authors. Published by Elsevier Ltd.

Selection and/or p eer-review un der responsibility of Dr. Manuel Fischer

Keywords: centrality; communities; modularity; interlocking directorates

ELSEVIER

1. Introduction

The goal of this paper is to explore the interlocking directorate network among companies listed on the French Stock Exchange by using social network analysis which focuses on the structure of the network. In particular, we concentrate our study on the French companies listed in the French Financial Index CAC40. Companies are selected from French financial market because of the availability of the data.

1877-0428 © 2013 The Authors. Published by Elsevier Ltd.

Selection and/or peer-review under responsibility of Dr. Manuel Fischer

doi:10.1016/j.sbspro.2013.05.058

Social networks (which were initially investigated by sociologists) have been of great importance in economics (Goyal, 2007; Slikker & van den Nouweland, 2001; Demange & Wooders, 2005). Networks are ubiquitous in social and economic phenomena. The use of methods from graph theory has allowed economic network theory to improve our understanding of many economic phenomena in which the embeddedness of individuals in their social inter-relations cannot be neglected. Social networks are formally defined as a set of actors or network members who are tied by one or more type of relations (Marin, 2010). There could also be many different types of relationships, such as collaborations, friendships, web links, citations, information flow, etc. (Marin, 2010). These relations represented by the edges in the network, are connecting the actors and may have direction (showing the flow from one actor to the other) and strength (showing how much, how often, how important the tie is).

Analysts of economic organization have displayed a long and continuing interest in intercorporate relationships. One form of intercorporate relationship that has received a great analytical attention is directorship interlocks. A director can belong to several directorships in different firms. Such a director constitutes a link between the related firms. We say that firms that are linked in this way are interlocked. There is much research on interlocks, ranging from a description of what the network of interlocked firms looks like to studies on the influence of interlocks on firm strategy and performance. Interlocking directorates create relational spaces where information and strategies may not be only exchanged but also co-defined and co-created. In other words, interlocks act as communication channels enabling information to be shared between boards via common directors. Moreover, the network of the board interlocks constitutes an important organizational resource for the firm (Bazerman & Schoorman, 1983). Early studies on the evolution of interlocking directorates mostly relied on firm covariates and considered the formation of a board tie as purely dyadic event. It is often shown that groups belonging to the same community are suitable to share common properties or play a similar role.

The question we focused on in this paper is whether a community structure exists among the French interlocking networks. Communities are cohesive subgroups of actors among whom there are relatively strong, direct, intense or frequent ties (Wasserman & Faust, 1994). Community structures are quite common in real networks. Social networks often include community groups based on common location, interests, occupation, etc. Detecting these communities is an important condition to understand the structure of the data and consequently the structure of the network. Researchers have been more active in exploring community in networks (Newman & Girvan, 2004; Newman, 2006; Fortunato, 2010). In fact, extracting communities allow us to discern important outlines and to capture fundamental properties of the network. A community is roughly defined as "densely connected" individuals that are "loosely connected" to others. Interlocking directorates became a favorite topic for several reasons. First, interlocking directorates provide information to the central organizations. Second, interlocks provide means for communication. Third it provides a means of signaling reputation to other firms.

We define social networks as information networks that are represented by graphs and illustrate the interactions between individuals or entities. In these networks, each individual is represented by a node in the network, and there is an edge between two nodes if an interaction occurs or a relationship exists between the two individuals during the observation time. In our case, we start by constructing a graph representing listed companies (vertices) and their connections (weighted edges for valued grapha and non weighted edges for non valuedb graph). This graph corresponds to the French intercorporate network.

In a previous work, Elouaer-Mrizak (2012) calculated different vertex centrality measures: degree, betweenness, closeness. The use of betweenness and closeness is most powerful to have an idea about the intensity of the relationship among companies. Elouaer-Mriz^ (2012) findings 'show that: first, financial institutions and big companies are central actors. Second, small companies tend to be connected to big companies in order to reduce environmental uncertainty, which is in line with Schoorman et al. (1981). Third, firms tend to have a banker in their boards (good signal), which is consistent with Davis and Mizruchi (1999) and Pfeffer and

a For valued graph, cells in the adjacency matrix are the number of shared directors between two boards.

b For non valued graph, cells in the adjacency matrix are equal to one if there is at least one shared director between the two boards, and equal to zero if there is no common director between the two boards.

Salanick (1978) findings. Moreover, Banks tend to have links with firms in order to reduce information asymmetry and to monitor the company (through dominant position in the board). It is certain that the centrality and position of the companies in the French Intercorporate network is important. Therefore, visualization of the network reveals the presence of small number of highly connected nodes (hubs) and community structure is evident.

In this work, we analyze the community structure of a real world financial network, namely intercorporate network or board network at three observation points 1996, 2005 and 2010 and identify the community structures in it. To do that, we use the modularity optimization (Newman & Girvan, 2004) method which detects communities by searching over possible divisions of a network for one or more that have particularly high modularity. This approach is one of the most popular functionsc; it attempts to measure how well a given partition of a network compartmentalizes its communities (Newman, 2004; 2006).

The main contribution of our paper is to find that French board network displays a strong community structure. The outline of the paper is as follows. In section II, we present hypothesis to test, our data and the methodology of our work. Then in section III, we present the board network in the French context. In section IV, we present the analysis of the community structure and economic interpretation of the results is given. Finally, we provide some concluding remarks in section V.

2. Hypothesis, Data and Method

2.1 Hypotheses

In our case, community detection is a task of detecting cohesive groups within the French intercorporate network. In addition to information about board composition, we have additional information about firms (nodes): type of organization and sectors they are operating in.

When we have data about node attributes, it might be relevant to extract group of nodes that are not only connected in the intercorporate network but also present some particular attributes. Incorporating attribute data in detecting communities has not been thoroughly studied yet in the context of interlocking directorates. In order to check whether the attribute data determine the composition of detected communities, we will test two hypotheses derived from the literature.

Hypothesis 1: Communities are expected to be composed of groups of industrial companies and financial firms.

In the interlocking directorates' literature, we find several motives that may explain the existence of links between firms. For example, researchers like Pfeffer and Salancik (1978) considered resource dependency as the greatest justification for interlocking among boards. This theory argues that interlocks are established to reduce uncertainty. Many researchers (like Burt, 1980; Burt, 1983; Boyd, 1990; Lang and Lockhart, 1990) justified that a firm creates a connection through an interlock to guarantee access to external resource. Consequently, uncertainty is reduced. Therefore, interlocks are viewed as a transfer device. It has been complicated to validate resource dependency because this view proposes that reducing uncertainty will raise profits. The firm's mutual profits will be higher if an interlock offers additional information to a company.

Based on interviews with bankers, Mizruchi (1996) and Richardson (1987) suggest that generally bankers sit on a board of a distressed firm (i.e. firms in financial difficulty). From the side of the distressed firm, it seeks additional interactions to obtain funds. Indeed, due to information asymmetries, banks have access to only firm's external information, but have less awareness about the quality of the debtors. Therefore they will try to find means to reduce information asymmetry. According to Mariolis (1975), interlocking directorates is one of the

c There are many computer algorithms to extract communities from a network data: modularity-based methods; spectral algorithms, dynamic algorithms and methods based on statistical inference (Porter, Onnela and Mucha, 2009).

ways that can help to reduce asymmetries of information. Having a board connection between industrial companies and financial firms allow banks to monitor debtors because they have access to internal information. Through these connections, financial f^s can keep the firm's management under their influence.

Consequently, financial institutions are a central actor in the interlocking network. Resource dependency theory states that interlocks diminish improbability while financial control theory asserts that access to funds increase the greatest concern.

Mintz and Schwartz (1985) consider the financial control theory as a stem of the resource dependence model. The grouping of these two theories would imply that interlocks take place more often between industrial companies and financial firms in particular banks. This provides industrial firms the ability of obtaining funds when required. Moreover, better information flows between borrowers and lenders. Following the discussion above, we can expect that communities will be composed of groups of industrial companies and financial firms.

Hypothesis 2: Communities are expected to be composed of groups of organizations engaged into similar topics, and operating in same sector.

The Clayton act of 1914 prohibited interlocking directorates among competing corporations but did not condemn the practice in general. The benefit of horizontal coordination between competitors may also be achieved through an indirect interlock. Horizontally interlocked firms can get advantages through communication regarding pricing, advertising, and research and development (Pennings,1977; Pfeffer, 1972) particularly in highly concentrated industries. However, these benefits are gained at the cost of the collective advantages of open market competition. It is justifiable to ask whether interlocks between competitors really make collusion easier. Studies of the U.S corporations by Pennings (1980) examined the association between industry concentration and horizontal ties. He found a positive association between the two. Moreover, Pennings (1980) examined whether such ties develop firm performance. He found a positive association between industry concentration and horizontal ties (interlocking directors between firms operating in the same sector). In the perspective of market for corporate control, Cotter et al. (1997) study interlocked directorates between bidder and target firms. Their findings suggest that the presence of directors' interlocks reduces the gains to target shareholders and decreases the likelihood that a target firm receives multiple proposals. In contrast, Barucci (2006) examines the hypothesis of collusion finding that interlocking directorates improve the ability of the controlling shareholders to expropriate the minority shareholders, and extract benefits from control.

Mizruchi (1996) concluded that the fact that within-industry interlocks continue to occur suggests that some interlocks may have been established with the aim of restricting competition. Interlocks are a mean for firms to exchange knowledge and strategy (Useem, 1984; Lorsch & MacIver, 1989; Haunschild & Beckman, 1998; Carpenter & Westphal, 2001).

Following the arguments above, we can expect that communities will be composed of groups of organizations engaged into similar topics and operating in the same sector.

2.2 Data

To identify community structure in the network of corporate boards, we use data from France. The French data are supplied by Dafsaliens. The data set for this study comprises lists of directors for the CAC 40 companies in France in 1996, 2005 and 2010.

Our data shows two changes occurring between 1996 - 2005 and 2005 - 2010. The first is the size of the board. In 1996, the average board size is 13.59 members per board, while in 2010; it is near to 18 members per board. This may be the consequence of the setting of the Dualist system of governance (Directoire et Conseil de Surveillance) which implies a large size of boards (it may attempts 22 members).

Table 1: Top 40 French companies and directors, 1996, 2005 and 2010

CAC 40

1996 2005 2010

No. of Companies 39 40 40

N. of Directors 530 631 610

Average Board Size 13.59 15.77 18.125

No. of Positions (board seats) 692 773 725

No. of Positions (seats) per director 1.30 1.22 1.19

Moreover, the number of positions per person decreases for CAC 40 companies between 1996 and 2010. Multiple directorships create "links and interlocks". Table 2 shows the number of directors who hold more than one position.

Table 2: Number of directors/positions for Top 40 French Companies

CAC 40

1996 2005 2010

1 442 535 522

2 48 60 69

3 21 28 14

4 11 6 3

5 5 2 1

6 1 0 1

7 1 0 0

>81 0 0

In table 2, we note that for CAC 40 companies, the number of directors holding two positions increases across time. The number of directors holding three positions increases from 1996 to 2005, and then decreases from 2005 to 2010. In general, the number of directors holding 5 or more positions decreased from 1996 to 2010. These multiple directorships create links between boards. Consequently, a network between boards is created.

2.3 Methodology

Networks

Affiliation networks are composed of two modes. The first mode is the set of agents N. The second mode is the set of events M. The number of agents in the network is n and the number of events is m. Interlocking directorates can be seen as an affiliation network. The boards of directors constitute the set of agents and the companies constitute the set of events.

We can represent an affiliation network by an affiliation matrix A of size nxm. We denote an element of A by atj is an element of A such that i is an element of i — 1,2,....,N and j is an element of j — 1,2, ....,M. intakes 1 if a director i sits in the board j.

_fl if i participates in j n)

aii "I 0 otherwise

Each row of A gives all the events in which an agent is taking part. The sum of a row of the affiliation matrix gives the number of events in which an agent is involved. Each column gives all the directors in a board.

Agents and events can be represented separately. More precisely, each mode can be represented as a network by using the other mode to define the relationships. With this approach, we observe only one mode at a time. In terms of graph theory, an affiliation network is represented by a uni-partite graph.

A network or a graph is described by a set of vertices (or nodes) and a set of pairs of vertices called edges which represent connections between vertices. Formally, a network or a graph may be denoted by G = (V, E) with V the vertex set and E the edge set which is a part of V xV. We denote by n the number of nodes in a graph and by m the number of edges or links. Graphically nodes are drawing as points and relationships as lines connecting pairs of points. A graph can be represented by a matrix A called the adjacency matrix. The adjacency

matrix has elements a equal to 1 if there is an edge from vertices i to y'and equal to 0 otherwise. Graphs can also

be valued or non-valued. A valued graph has numbers attached to the lines that indicate the strength or frequency or intensity or quantity of the tie between nodes.

If A is a symmetric matrix, the graph is undirected which means that the ties have no direction (i.e., i -> 7'is equivalenttoy -* i). Otherwise, in directed graphs (also known as digraphs), the ties have direction and these lines are called arcs.

Centrality is regarded as one of the most important and commonly used conceptual tools for exploring and measuring actor roles in social networks. A node's degree centrality, in an undirected graph, is defined as the number of nodes that are connected to that node. Thus the degree of a node i of an undirected graph is given by ki = c-ij = H; a-ji and the degree sequence is given by the following list {fc1( k2,—, kN] of the node degrees.

The degree distribution is provided by {P(fc)< k £ N} which specifies for each integer k the fraction of nodes such that fcj = k. P(fc)can be considered as the probability that a node has degree k. The average degree of the network is defined as the average of the degrees over all nodes in the network. However, average degree might not be representative, since the distribution of degrees might be skewed. For example, Bernoulli graphs of low density tend to have degree distributions with some positive skewness but without having very high degree nodes (hubs). Moreover, Bernouilli graphs of higher density tend to have symmetric degree distributions without skewness.

A graph is connected if it is possible to find a path from any vertex to any other vertex of the graph. A path is an alternating sequence of vertices and edges, starting and ending with a vertex, in which each vertex is incident with the edges following and preceding it in the sequence and the number of edges is called the length of the path.

Paths are useful to measure distance, i.e. how far apart vertices are in a graph. The shortest path between two vertices is referred to as a geodesic and its length is the geodesic distance (or simply the distance) between two nodes. The average geodesic distance between two vertices in a connected graph is the characteristic path length t. The maximum geodesic distance from vertex i to any other vertex is its eccentricity. The maximum eccentricity in a graph is its diameter.

An induced subgraph, or simply a subgraph of a graph G = (K, E) is a graph H = (V', £") whose vertex set V' is a part of V, and the edge set E' contains exactly the edges of G with both endpoints in V'.

The dyad consists of two vertices that are either adjacent, i.e. connected by an edge, or not. Sub graphs of size three are called triads. If the three vertices are connected, they constitute a triangle. A triangle is the smallest nontrivial example of a clique. Cliques are complete (or fully connected) subgraphs, in which all vertices are adjacent to each other. Cliques have the property that transitivity holds within the clique; i.e. if vertex i is a neighbor of vertex j and j is a neighbor of vertex h, h is also a neighbor of i.

Components are maximally connected subgraphs. If a network is not connected, the vertex set can be partitioned into components C1, C2, ...,Cm with ni>n2> ••• >nm> 0 nodes = n). Components are

connected and they are not part of a larger connected sub-network.

To detect communities in the network, we must consider each component separately. A subset Chs N is called a community if the density of links internal to Ch is much larger than the density of links connecting Ch to the rest of the network.

Community structure

In this paragraph, we introduce the basic terminology on communities and methods of community detectiond. The detection of community groups, or modules, within networks is one of the great current interests. Roughly speaking, a community group is a portion of the network whose members are more tightly linked than to other members of the network. A multitude of approaches (Clauset, Newman & Moore, 2004; Girvan & Newman, 2002; Goldshtein & Koganov, 2006; Hastings, 2006; Newman & Girvan, 2004; Reichardt & Bornholdt, 2006) have been considered to explore this concept.

Properties of the sub networks may differ from the aggregate properties of the network as a whole, e.g., modules in the World Wide Web are sets of topically related web pages. Thus, identification of community groups within a network is a first step towards understanding the homogeneous substructures of the network.

Methods for identifying community groups can be adapted to different types of networks (for bipartite networks for example, see Barber, 2007). These methods are relevant for our study of the boardroom networks, allowing us to examine the community structure in the board networks.

To identify communities, we consider the modularity, introduced by Newman and Girvan (2004). Newman and Girvan (2004) proposed the first community detection approach using the social network analysis techniques and opened a new setting for community detection algorithms. Their method is a divisive hierarchical clustering algorithm which iteratively removes the edge with highest betweenness to obtain the community structure of the network. The betweenness of an edge could be computed as the number of shortest paths running through that edge. High betweenness is a sign for bridges in the network, which are edges connecting different communities, illustrated in Figure 2.

Community structure

Fig. 1. Community Structure, Source: Newman (2004)

dSee Newman (2003), Newman and Girvan (2004), Boccaletti et al. (2006), Fortunato and Castellano (2009), Fortunato (2010).

Fig. 2. Edge Betweenness: edges connecting different communities have high betweenness. In this figure, the thickness of edges represents their betweenness. The edge between two communities has the highest betweenness as all the shortest paths between any pair of vertices, which are in different communities, have to run through this edge.

Modularity makes intuitive notions of community groups precise by comparing network edges to those of a null model. To identify communities within networks, there are different methods (Fortunato, 2010). The different methods analyze graph structure in different ways but in each case, the strength of the different graph partitions they suggest are evaluated by comparing the distribution edges with those expected in random or null model with the same structural characteristics (Modularity).

The modularity Q is proportional to the difference between the number of edges within communities Ch and those for a null model:

According Newman and Girvan (2004), and for a given network with vertex set V and a partition Clt C2,..., Cq (i.e., Ufc Ch = V and Ch n Ck = 0 for all h, k), the modularity is given by:

Q=^=^h[aij~k-^] (2)

The degree of nodes i andy'are kt and kj respectively. L is the number of links. Q is obtained by summing up, through all the sets Ch , the difference between the actual number of links internal to the set QXt; fly) and the

value expected if links were created randomly, regardless of the existing communities (the NULL Model) l k'k

2^-,ii~2Lj. Consequently, Q is large (it tends to 1, due to the normalization) when the density of links into the communities Ch is larger in comparison to a random distribution of links in the network.

The standard choice for the null model constrains the degree distribution for the vertices to match the degree distribution in the actual network. Random graph models of this sort are obtained (Chung and Lu, 2002) by putting an edge between vertices i and j at random, with the constraint that on average the degree of any vertex i is dt.

The null model choice is that Q = 0 when all vertices are in the same community. The goal is to find a division of the vertices into communities such that the modularity Q is maximal.

Finding the partition which maximizes the modularity is NP-hard (Brandes et al., 2006) and therefore many approximation algorithms have been proposed, see Fortunato (2010), Porter et al. (2009), Schaeffer (2007) for comprehensive surveys. In this paper, we will use, in addition to the edge betwenness algorithm (Newman and Girvan, 2004), two other different algorithms:

• The first one, Walktrap (Pons & Latapy, 2006), is based on the idea that a small random walk will stay inside the community from where it's originating because there are many links inside and few bridges leading outside.

• The second one, Fast Greedy (Clauset et al., 2004), is a greedy optimization algorithm for modularity: each node is initially in its own community and then, at every step, the algorithm groups two communities in order to maximize the gain of modularity.

Evolution of Communities

One important element of our study is the availability of the data on three period of time (1996,2005 and 2010). To study the evolution of these communities over time, it is important to define different forms of transition of these communities (Sitaram Asur et al., 2001)

Continuation: A community continues from one period to a next period (period=1996, 2005,2010), if it remains same from one time to another. We do not impose the constraint that the edges sets should be the same. The central motivation behind this is that if certain nodes are always part of the same community, any information supplied to one node will eventually reach the others.

Dissolution: A community is said to have dissolved if none of the vertices in the cluster are in the same cluster in the next period. Logically, a dissolve indicates the lack of contact or interactions between some nodes in a particular time period.

Formation: A new community is said to have been formed if none of the nodes in the community were grouped together at the previous period. Intuitively, a formation indicates the creation of a new community.

Merging: Two different communities are marked as merged if there exist a community in the next period that contains at least k%e of the nodes belonging to these two clusters.

Intuitively, it implies that new interactions have been created between nodes which previously were part of different communities. This caused k% of nodes in the two original communities to join the new community. Note that, in an ideal or complete merge, with k = 100, all nodes in the two original clusters are found in the same community in the next timestamp.

Splitting: A single community is marked as split if k% of nodes from this cluster, are present in 2 different clusters in the next period. Intuitively, a split signifies that the interactions between certain nodes are broken and not carried over to the current period, causing the nodes to part ways and join different communities.

3. The French Board Network

3.1 The Network's characteristics

Multiple directorships create "links and interlocks". Table 1 above shows the number of directors who hold more than one position. In other words, interlocking directorates results from relationships between firms' boards. Interlocking directorates can be defined as companies that "interlock" their boards by common or shared directors. An important question arises: How should one draw a graph to represent the network resulting from common directors? Indeed, there are two units of analysis either boards or directors. One could treat the "boards" as the basic unit of the analysis and form a graph whose vertices represent boards and whose edges represent shared directors (board graph). Alternatively, one could make a graph whose vertices represent directors and whose edges represent shared board memberships (director graph). There is no obvious way to make a choice among them. Theref are two sorts of social entities here, the directors and the boards, and the network's edges represent membership of the former in the latter.

In a bipartite network, directors and boards correspond to two sets respectively ND and NB and a link between two nodes i and y exists when the director iEND sits in the board j 6 NB. Board network and director network are projections of a bipartite network.

In this section, we analyze the French corporate board network for CAC40 companies listed in the French Stock Market in three years: 1996, 2005 and 2010.

e As Sitaram Asur et al. (2001), we used k values of 30 and 50 in our case.

f For an excellent discussion of graphs, their representation, manipulation and application to the social sciences, see Wasserman and Faust (1994)

In this paragraph, we are interested in the network characteristics (Table 3) for better understanding connections and its mechanism of formation. The basic properties which characterize connectivity within a network are connected to the concept of degree.

Table 3. Statistics on the two mode network of boards and directors for CAC40 companies

1996 2005 2010

Boards Directors Boards Directors Boards Directors

Number of Nodes 40 530 40 625 40 610

Average degree 17,225 1,3 19,2 1,229 18,125 1,189

Min degree 1 1 12 1 11 1

Max degree 36 9 26 5 28 6

The projection of the two mode network on the set of boards generates the board network, which is fully described by the symmetric 40 X 40 connectivity matrix. The CAC40 board network has a big component in 1996. In 2005 and 2010, the graph is connected. In 1996, the network is composed by 5 components one of them is a giant component as it has 34 nodes which is 85% of the total. The other components are isolate vertices.

Table 4. Global metrics for the CAC40 board network

1996 2005 2010

Components 5 1 1

Isolates 4 0 0

Density 0. 2470 0.2615 0.1974

Degree Centralization 0. 2667 0.3441 0.0830

Betweeness Centrality 0. 0944 0.0963 0.0927

Clustering Coefficient 0. 590 0.343 0.387

Figures 2, 3 and 4 present the degree distribution of the networks. For 1996, we limit our attention to the giant component.

Table 5. Statistics on the CAC 40 board network

1996 2005 2010

Boards Directors Boards Directors Boards Directors

Number of Nodes 40 491 40 625 40 610

Average degree 11,35 26,24 8,15 23,18 6,4 21,38

Min degree 1 7 1 11 1 10

Max degree 20 173 19 100 16 105

Average distance 1,788 2,516 1,90 2,829 2,244 3,106

Diameter 5 5 4 5 5 5

The diameter is the maximum of the shortest path distances between two connected nodes of the network. It is the maximum distance between two connected nodes. If the diameter is 1, each node considered is directly related to all the others. If it is of 2, one needs a maximum of 1 intermediary to contact the other nodes of the network, etc.

If the diameter is small, then each node can be connected to any other by a short chain of social relations: it is a "small world8". On the contrary, if the diameter is large, the nodes are very distant from each other. A diameter of 1 corresponds to a complete network. A second way of detecting the effect of small world would be to calculate the average of the geodesic distances'1 or minimal distances:

l = I^-¿izidij 3

Where gy is the geodesic distance between nodes i and j and N is the number of nodes. If the network is a "small world" then I s Log(N).

Table 4 shows that the diameter is weak in CAC40 board network. It is quite constant and equal to 5 for the most point of times (1996, 2005 and 2010). Boards are never separated by more than 4 intermediaries in the board network. To illustrate, for example, Enron directors were sitting on the boards of 10 others of the Fortune 1000 American companies, including Compaq, Eli Lilly, Lockheed Martin, and Motorola. The administrators of these ten boards were participating in boards of 49 other Fortune 1000 companies. From Enron, 648 boards could be achieved in only four maximum degrees of separation. In terms of people, 95 administrators were directly related to the Enron Directors through boards of directors, and 482 others were far from two arcs of the Enron Board.

For the CAC40 board network, the average of the geodesic distances varies from 1.788 to 2.244. And for the CAC40 director network, the average of the geodesic distances varies from 2.516 to 3.106.

3.2 The node degree distribution

The degree of a node in a network is the number of connections or edges the node has to other nodes:

fc|=Zay (4)

Where ai;equals 1 if there is a connection between nodes i and j, and equals 0 otherwise. The node degrees distributions for the 40 firms (Figures 3, 4, 5) show some heterogeneity between nodes: some firms with a great number of common directors and some with only some connections. The distribution does not follow a normal one. Thus, the networks formation is not purely random. Links between companies are not random; they are the outcome of a complex interaction among firms.

g One speaks about .small world.(small world) when each individual of the network can be connected to any other individual by a short chain of social relations including at maximum 5 other nodes (or six degrees of separations). It is Milgram (1967) who put the assumption of the small world. He had asked 296 people to forward a .le to a target person by using only intermediaries who knew each other. The targeted person was a stockbroker residing at Boston (known information of the participants). At the end of the game, 64 files arrived at the objective individual. The remainder corresponds to chains made incomplete by the abandonment of a participant. The average length of the chains is 5.2 intermediaries. This small number of intermediaries excited the curiosity of the researchers who formulated the assumption of "small world". With six intermediaries, one can connect two people who do not know each other and thus reach anyone in the world.

h The geodesic indicates the shortest way, or one of the shortest ways if there are some several, between two points of a space equipped with metric.

Fig. 3. Degree distribution 96

Fig. 4. Degree distribution 05

Fig. 5. Degree distribution 10

3.3 Communities in the network

A community is a set of nodes such that the density of internal connections is stronger than the density of connections towards outside. The goal is thus to find a partition Cv C2 —,Ck of nodes in communities checking this definition. The visualization of the graph allows us to detect groups by characteristic. Community analysis applied to the networks (for 1996, we consider the giant component) shows a quite strong community structure. Table 5 presents the number of communities for each network. In order to calculate, Q values, we used three methods: edge betweenness, walk trap and fast greedy algorithm.

Table 6. Number of communities for each network

1996 2005 2010

Edgebetweenness Number of communities 16 11 14

Modularity (Best Split) 0.02139615 0.145581 0.1836524

Walktrap Number of communities 4 10 8

Modularity (Best Split) 0.2431313 0.2161188 0.2726851

Fastgreedy algorithm Number of communities 3 5 4

Modularity (Best Split) 0.2656191 0.2985871 0.3221032

Recall that modularity measures the strength of the determined communities based on the relative number of intra-group connections versus extra-group connections. We observe that in 1996, modularity has a low value (0.0214). Good partitions give large positive values of Q. In fact, the way the modularity was defined (as it has been normalized), Q ranges between —1 and 1. Q — 0 means that all vertices of the graph are assigned to the

same community. At the other extreme, if we have a partition with each community being a singleton, then Q turns out to be negative. In particular, if for any partition of a graph, modularity is found out to be non-positive, then this graph has no community structure (in fact, such a graph would show a strong multipartite structure, in the sense that it would be decomposed to certain subgraphs with very few internal edges and many edges lying between them). If modularity takes large positive values (close to 1), then this indicates that the graph is decomposed to communities i.e. more edges in the graph within these communities than what would be expected by chance.

When edge betweenness algorithm considered, values of modularity are increasing across time among the CAC40 board companies (from 0.0214 in 1996 to 0.1837 in 2010). In 1996, almost all the boards are allocated to the same communities. However, in 2005, the modularity value is medium. This means that the structure of the board network among French companies is not random, and communities start to emerge. According to hypotheses 1 and 2, communities are expected to be composed of groups of organizations operating in the same sector and composed of groups of industrial and financial firms. Our results confirm these two hypotheses. To illustrate these findings, we examine the communities in each year (see appendix 1) we see that, for example: in the community {Société Générale, Total, Saint Gobain, PPR}, we find that Société Générale and PPR are in the same community. In fact Société Générale holds 6.86% in the capital of Total. Moreover, Michelin and Renault are in the same community. Michelin and Renault both are industrial companies and are operating in two complementary industries.

Community structures identify the existence of forms of coalitions among CAC40 companies. Control is one of the most extreme forms of influence, which is the power to maneuver the decisions of a company towards a direction. The board network represents the connections among companies (or directors). Directors are the actors who, via co-membership on boards, interact and communicate with one another. Corporate board meetings are the events that bring directors together, into face-to-face contact. Collective decision making of directors renders the corporate board an actor in its own. Directors also speak on behalf of the corporation to other corporations, and, on occasion, to the public as well. CEOs will often serve on the boards of other corporations, in part as a matter of prestige, as this illustrates that their advice is valued outside of their home corporation. Also, this is a way of obtaining valuable, and perhaps sensitive and privileged, business intelligence, and as a means of extending their own social networks.

The network analysis gave us abundant information on the network of interlocking directorates. The positions of banks in the network and the relations between bank and industry are significant. The finance capital theory expects that banks are involved in strong and multiple interlocks. In 2010, we obtain 14 communities (according to the edgebetweenness criteria). Among them we find: {Alstom, Axa, BNPParibas, Bouygues, Carrefour, Eads, Lafarge, Lagardere}; {PPR, Société Générale, Saint Gobain, Total}; {Alcatel Lucent; Cap Gemini; Crédit Agricole}. Interestingly, the main French banks (BNP Paribas, Société Générale, Crédit Agricole) are in the three big communities which is in line with Elouaer-^ra^ (2012) findings' that states thirty percent of the most connected firms are financial institutions. In fact, according to Dooley (1996), financial interlocks occur for several reasons. First, companies that are in financial difficulty tend to form a close association with one or more financial houses. Second, banks find it advantageous to be connected with large firms through electing company officers to the bank's board of directors as this may attract large deposits as well as secure a reliable customer for bank loans. Third, these financial interlocks also arise from the trust operations of banks. Moreover, the location of a firm in a network directly impacts access to information. Centrality in a network can be seen as the importance of a firm in that network. Betweenness measure may provide an idea about the importance of a company in the network. Boards that are not directly connected might depend on another board if it lies on a path connecting them. If a board lies on many paths connecting different components in a network then it has a high betweenness centrality. From this side, we can suppose that agents among communities are looking for intermediary positions. However, modularity values derived with the edge betweeness values are not high in comparison with walk trap and the fast greedy algorithms. Finally we can conclude that the French board network shows a quite strong community structure.

One another contribution of our work consists in exploring the evolution of the community structure of the French intercorporate network from 1996 to 2010.

The first point is that in 1996, we find two big communities of 9 actors. In contrast, in 2005 and 2010, we had, at one hand a split of these two big clusters and on the other hand, there is a creation of other communities. It is important to note that in the communities and from 1996 to 2010, there are groups of nodes that belong to the same community {PPR+Société Générale + Saint Gobain+Total} and {BNP Paribas+Bouygues+Lagardère}. Consequently, we can conclude that we have a strong cohesion between these nodes. In 1996, we find {Accor+Air Liquide+Alcatel+Aventis+Bouygues+Compagnie Bancaire+Lagardère+LVMH+Paribas}. On May 2000, Compagnie Bancaire and BNP merged together and we obtain BNP Paribas. Moreover, in 1996 we had 14 nodes that are not involved in any community. In next periods, 2005 and 2010, these nodes appear: indeed they belong to a community but were not present in any community in the previous period (1996). This simple event indicates the introduction of a company to a network.

In contrast, in 2005 and 2010, several nodes disappear. It means that in 1996 they were when they are found in a community but are not present in any community in the period. This indicates the departure of a company from a network.

To conclude, we remark that the main French banks continue to exist with the same nodes over the three periods. Our findings illustrate the important role of banks in the French intercorporate network among CAC 40 companies. As mentioned previously, interlocks between banks and industrial companies are considered in the bank-control model. These links may guarantee to the lenders a certain control over the companies in which they invest (bank control thesis; Scott 1985 cited in Hopner et al., 2003). We remark that the main French banks continue to exist with the same nodes over the three periods.

4. Conclusion

French board networks reveal particular properties in comparison to those in other countries. According to Windolf (2002), among the 374 main firms in France, fewer are interconnected; isolates represent a bigger share (43%) than in Germany (32%), and mostly than in the United States (14%) and in the United Kingdom (8%). However interconnected firms are more integrated than the other countries. The share of firms with ties is higher (4.92) than in Germany (4.21), in the United States (1.89) and in the United Kingdom (1.53). These firms do not only exhibit more interlocks, but also more multiple interlocks among each other. The proportion of multiple relationships (20%) is rather high. Moreover interlocks in France seem to be more centralized, as the firm with the highest number of interlocks in these countries is French. In brief, the French corporate network has the most cohesive core (Comet & Pizzaro, 2011). Consequently, extracting communities in the French board network is interesting.

To the best of our knowledge, this paper represents the first effort to examine the community structure among interlocking directorates of French companies. Our results contribute to the growing literature on interlocking directorates by showing that intercorporate network among CAC40 French companies reveal a structure with communities. Several facts may give this result, such control and environmental uncertainty.

Overlap in group membership allows for the flow of information between groups, and perhaps coordination of the group's actions. According to Sonquist and Koenig (1975), interlocking among boards through common directors, make the coordination between firms easier. Interlocks allow firms and/or financial institutions to reduce dependence on formal company communications, and/or co-opt, control, and monitor other firms (Haunschild & Beckman, 1998). Interlocks facilitate to firms obtaining valuable information about other firms, as the primary function of interlocks is to manage dependence through cooptation and control (Haunschild & Beckman, 1998). Interlocking directorates can therefore be seen as devices of power and influence for one company over another.

Our findings are in line with these facts since we find main financial institutions among the large extracted communities. Moreover, diversification strategies are often associated with control and coordination problems.

The purpose of horizontal interlocking would be to coordinate firms within the group. Horizontal interlocks contribute to maintain and promote transactions between group members, and create a communication network.

As a further research direction, we should extend our work to companies listed in the French financial index SBF 250 which includes 250 companies and compare the two networks with the ownership networks. Moreover, we should study the evolution of our communities in more details as in Sitaram Asur et al. (2001).

Acknowledgements

We would like to thank Jean Benoît Zimmerman and Cilem Selin Hazir for their helpful comments. We wish to acknowledge the participants of the 9th International conference on Applications of Social Network Analysis for their comments during the session. We are grateful to the two anonymous referees for their valuable comments and constructive suggestions on this paper.

References

Barabasi, A.-L. (2003). Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life. Plume Books.

Barber, M.J. (2007). Modularity and community detection in bipartite networks. Physical Review 76, 66-102.

Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. (2006). Complex networks: Structure and dynamics. Physics Reports, 424, 175-308.

Brandes, U. D. Delling, M. Gaertler, R. Goerke, M. Hoefer, Z. Nikoloski, and D. Wagner (2006), "Maximizing Modularity is haid,"ArXiv Physics e-prints.

Chung, F. and Lu. L. (2002). Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6(2):125-145.

Clauset, A., Newman, M. E. J., and Moore, C. (2004). Finding community structure in very large networks. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 70(6):66-111.

Comet, C. and Pizzaro, N. (2011). The cohesion of intercorporate networks in France. Procedia Social and Behavioral Science. 10 (2011) 52-61.

Cotter, J., A. Shivdasani, and Zenner, M. (1997). Do Independent Directors Enhance Target Shareholder Wealth During Tender Offers?,

Journal of Financial Economics, 43, p. 195— 218.

Davis, GF & Mizruchi, MS 1999, "The money center cannot hold: Commercial banks in the US system of corporate governance',

Administrative Science Quarterly, vol. 44, 215-239.

Demange, G. and Wooders, M. H. (2005). Group Formation in Economics: Networks, Clubs, and Coalitions. Cambridge University Press. Dooley, P.C. (1969). The interlocking directorate. American Economic Review, 59, 314-323.

Elouaer-Mrizak, S. (2012). A social network analysis among French companies. Proceedings IEEE/ACM International Conference on

Advances in Social Networks Analysis and Mining, 2012, 1018-1026.

Fortunato, S. (2010). Community detection in graphs. Physics Reports. 486 (3-5) 75-174.

Fortunato, S. and Castellano, C. (2009). Community structure in graphs, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and System Science, Springer-Verlag, Berlin, 2009, 1141-1163.

Gladwell, M. (2002). The Tipping Point: How Little Things Can Make a Big Difference. Abacus.

Gol'dshtein, V. and Koganov, G. A. (2006). An indicator for community structure. Preprint, July 2006. URL

http://arxiv.org/abs/physics/0607159.

Goyal, S. (2007). Connections: An Introduction to the Economics of Networks. Princeton University Press.

Hastings, M. B. (2006) Community detection as an inference problem. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 74(3):035102. doi: 10.1103/PhysRevE.74.035102. URL http://arxiv.org/abs/cond-mat/0604429.

Hopner, M. and Krempel, L. (2003). The Politics of the German Company Network. MPIfG Working Paper 9. http://www.mpifg.de/pu/workpap/wp03 -9/wp03-9.html

Lang, J., Lockhart, D., (1990). Increased environmental uncertainty and changes in board linkage patterns. Academic Management Journal 33, 106-128.

Girvan, M., and Newman. M. E. J. (2002). Community structure in social and biological networks. PNAS, 99(12):7821-7826. Newman, M. (2006). Finding community structure in networks using the eigenvectors of matrices. Physical Review E-( Statistical, Nonlinear and Soft Matter Physics), 74.

Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167-256.

Newman, M., and Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E, 69(2).

Pfeffer, J. and Salancik, G. R. (1978). The external control of organizations: A resource dependence perspective, Harper & Row, New

Piccardi, C., Calatroni, L. and Bertoni, F. (2010) Communities in Italian corporate networks. PhysicaA, 389(22), 5247-5258. Pons, P., and M. Latapy, (2006) "Computing communities in large networks using random walks," Journal of Graph Algorithms and Applications, vol. 10, pp. 191-218.

Porter, M. A., Onnela, J-P. and Mucha, P. J., (2009). Communities in networks. Notices of the AMS, 56 (9), 1082-1166. Porter, M. A., P. J. Mucha, and J.-p. Onnela, (2009) "Communities in Networks," Notices of the American Mathematical Society, vol. 569.

Reichardt, J. and Bornholdt, S. (2006). Statistical mechanics of community detection. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 74(1):016110, 2006. URL http://link.aps.org/abstract/PRE/v74/e016110. Schaeffer, S. E., (2007) "Graph clustering," Computer Science Review, vol. 1, pp. 27-64.

Schoorman F., Bazerman M., Atkin R. (1981). Interlocking Directorates: a Strategy for Reducing Environmental Uncertainty. Academy of Management Review, 6, 243-251.

Slikker, M. and van den Nouweland, A. (2001). Social and Economic Networks in Cooperative Game Theory. Springer.

Sonquist, J., Koenig, T., (1975). Interlocking directorates in the top US corporations. Insurgent Sociologist, 5, 196-229.

Wasserman, S., Faust, K., and Iacobucci, D. 1994. Social Network Analysis : Methods and Applications (Structural Analysis in the Social

Sciences). Cambridge University Press.

Windolf, P. (2002). Corporate networks in Europe and the United States. New York: Oxford University Press.

Appendix

A.1. Communities in CAC 40 board network in 1996