Modelling the evolution of a bipartite network—Peer referral in interlocking directorates☆
Introduction
A bipartite network defines a graph on a node set that represents two distinct types of social entities and where there may be relational ties only between nodes that are of different types. The social entities may be individuals and social clubs, authors and journal articles, directors and corporate boards, etc. Since bipartite networks make the conceptual link between social actors and social groups explicit, the study of bipartite networks is of great theoretical importance for understanding the duality of persons and groups (Breiger, 1974, Doreian, 1979, Feld, 1981, Freeman and White, 1993, Pattison and Robins, 2002). The nodes and ties in bipartite networks may be researchers citing papers (e.g., Small, 1973); people participating in social events (Breiger, 1974); partners in crime (Frank and Carrington, 2007); or directors on corporate boards (Mizruchi, 1996). This paper is focused on the last case, directors and boards. A so-called interlocking directorate is a bipartite board-to-director network consisting of vertices representing corporate boards and directors. A tie exists between board i and director j if j is a member of board i.
Most previous research on bipartite networks has focused on one of the two unimodal networks where two nodes of one type are considered tied if they share at least one alter of the other type. The obvious benefit is that methods developed for the analysis of unimodal networks applies. But as pointed out by Robins and Alexander (2004), the duality in defining for example the individuals by their affiliations with certain events or in defining the events by their ties to individuals, makes it hard to give priority to one type of node over another (Breiger, 1974, Galaskiewicz et al., 1985, Breiger and Pattison, 1986). Transforming a bipartite network to a unimodal network always means a loss of information (Borgatti and Everett, 1997). In particular, the strength of a tie is lost if we ignore the number of nodes of one type that two nodes of another type share. It can also be shown that a completely random bipartite network may give rise to quite interesting but altogether spurious structural features in the unimodal representations (see also Newman et al., 2001).
Because of the inherent dynamic nature of social networks, a wide variety of statistical models have been proposed for studying network evolution over time. While several methods assume that changes are made in discrete steps from one moment to the next (Katz and Proctor, 1959, Wasserman and Iacobucci, 1988, Sanil et al., 1995, Robins and Pattison, 2001), considerable advantages may be had from modelling longitudinal social networks in continuous-time (cf Snijders, 1996, Snijders, 2001). The early models for longitudinal social networks using continuous-time Markov chains (Holland and Leinhardt, 1977a, Holland and Leinhardt, 1977b, Wasserman, 1980a, Wasserman, 1980b, Leenders, 1995) built on the assumption that the dyads evolved independently of each other, something which excludes the exploration of dependencies on larger structures than the dyad. In response to this Snijders proposed a flexible and empirically testable class of stochastic actor-oriented (or actor-based) models that draw on the assumption that actors strive towards organising their social ties in a utility maximizing manner (Snijders and van Duijn, 1997, Snijders, 2001, Snijders, 2005, Snijders, 2006).
However, so far the evolutionary network models have not been adapted to the study of the evolution of bipartite networks, although there are some longitudinal studies of the evolution of interlocking directorates. Galaskiewicz and Wasserman (1981) fit a discrete-time model to the unimodal corporate network implied by interlocks in a small bipartite network. Like previously proposed models it also assumed dyad-independence. In their study they focused primarily on the corporations and they distinguished between different industries and used a qualified version of board membership that gives directed interlocks. Mizruchi and Stearns (1988) model the appointment of financial directors on non-financial boards for a small selection of boards. Studying the dichotomous event “hiring” or analysing the evolution of the induced unimodal network, prevents investigation of the strength of interlocks. Similarly, in these two models changes are largely explained through corporate specific covariates (size, solvency, profitability, etc.) rather than in terms of structural aspects of the bipartite (and unimodal) network(s), i.e. endogenous self-organising principles.
Here we propose a model for studying the evolution of bipartite networks that draws on Snijders’ work on stochastic actor-oriented networks. We analyse data on corporate boards for all firms traded on the Stockholm Stock Exchange's primary list between 1996 and 2005 (Edling and Sandell, 2001, Bohman, 2006). In our application, we focus on the processes of appointment to and departure of directors from boards based on a theory of action in the context of interlocking directorates. Central to this concept is the occurrence of multiple interlocks, which we elaborate on in Section 3.
We proceed by describing the statistical model. This is first done using a general model formulation with an inference strategy for the evolution of bipartite graphs that follows the model proposed in Koskinen and Edling (2004) that is in turn an extension of the stochastic actor-oriented models for directed unimodal graphs of Snijders (2001). We then discuss peer referral in interlocking directorates, in Section 3, before detailing the modifications needed for analysing interlocking directors in Section 4. We then fit models with structural features and with additional controls for director attributes in Section 5. To test the goodness of fit for the model, in Section 6 we compare forecasts from the model with observed data. We finish the paper with some concluding remarks.
Section snippets
Model formulation and estimation
We consider a bipartite graph on a fixed vertex set V that is the union of two disjoint subsets A and B, , . We denote by N the subset of pairs of vertices that constitute the set of possible relational ties between vertices. In the case of a bipartite graph , there can only be ties between vertices from different classes. The aim is to model the evolution through time of the random set of edges . The (generalised) adjacency matrix is a collection of
Interlocking directorates and peer referral—the action mechanism
This section presents an action mechanism for the actor-oriented model of network evolution. We specify and interpret the model based on two interacting mechanisms: peer referral and homosociality (or cognitive gender homophily). Before we discuss these mechanisms, we wish to briefly underscore the importance of explicitly considering the bipartite structure in the analysis of interlocking directorates.
One can easily transform a bipartite board-to-director network into two one-mode symmetric
Modelling and estimating peer referral in interlocking directorates
Before we define the specific forms of (1) and (2), given in Section 2, the longitudinal study of interlocking directorates requires a few further modifications of the model specification. These are modifications that stem both from the nature of interlocking directorates as well as from computational considerations that have to be taken into account when dealing with large datasets.2
Results
For setting initial parameter values and variance–covariance matrix for the proposal distribution we followed Koskinen and Snijders (2007) and obtained the Hessian of and parameter values that maximised the likelihood for arbitrarily chosen shortest paths. The Metropolis–Hastings algorithm was implemented with a total of 4000 iterations and a burn-in period of 1000 iterations.
Summaries for the posterior of the purely structural model are given in Table 2 and comparisons of interval estimates
Goodness of fit
We may use Bayesian forecasting to assess the goodness of fit of a stochastic actor-oriented model (Koskinen and Snijders, 2007). If the model is fit to observations and , a third observation , which is not used in fitting the model, may be used for model evaluation. The idea is to assess how probable is given and , unconditional on the parameters. The distribution of given and is unconditional on the parameters is the posterior predictive
Discussion
We introduced a model for bipartite evolution drawing on Snijders’ work on stochastic actor-oriented models, and showed how to apply this model for the study of interlocking directorates. Analysing data on firms traded at the Stockholm Stock Exchange between 1996 and 2005, we found that creating four-cycles seems desirable for corporate boards but that preserving them is not as important. We conclude that this is a sign of peer referral in the recruitment process, and that this is central to
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An earlier version of this paper was presented at the workshop on “The Interdependence of Networks and Behaviour Psychological, Sociological, Political and Economic Perspectives” in Melbourne in 2008, at which we received many useful suggestions. This work has benefitted from helpful comments and suggestions by Malcolm Alexander, Peng Wang, and Garry Robins. We gratefully acknowledge the financial support from the Swedish Council for Working Life and Social Research and from Riksbankens Jubileumsfond. The research of J. Koskinen was partly done while being a Research fellow at the School of Behavioural Science, University of Melbourne. Support is also acknowledged for part of the work of Koskinen from US National Institutes of Health (NIH 1R01HD052887-01A2).