Interlocking directorates in Irish companies using a latent space model for bipartite networks

Data.

A list of all Irish-based companies listed on the ISE from 2003 to 2013 was obtained from the Exchange. The company registration office number and address of each of the remaining companies was obtained from the Irish Companies Registration Office website. Hence, for each company, a list of their directors was found. The data collected were rearranged into a bipartite dynamic graph, whereby for each year edges between directors and companies were created according to boards’ compositions. The network was made up of 1,009 directors, 91 companies, and 4,952 edges. Because the main focus of our analysis is the study of director interlocks, we removed those companies that did not involve any interlocking links over the study period, in other words, those companies that did not share any of their directors with any other board over the 11 years of the study. The resulting dataset (corresponding to 761 directors, 59 companies, and 3,855 edges) was used to carry out our analysis. Only 22 of the companies were quoted on the ISE for the entire time period considered, whereas all other companies either joined at a later stage or left early or both. There was also a significant number of directors who sat only on one board, sometimes for only a year. These directors can be seen as peripheral to the Irish network of boards and directors. Table 1 gives an overview of the number of directors and boards in each year of the study.

Exploratory Analysis.

An interlocked director in any given year is a director who sits on two or more boards in that same year. It can be seen from Table 1 that the proportion of interlocked directors increased from 2006 to 2010 and declined from 2011 to 2013.

The number of interlocking links for a node is related to the excess degree, defined as follows: for a director with degree d, it is equal to d−1 if d>1 or 0 otherwise. Fig. 1 gives further evidence that the proportion of interlocking links increased over time until around 2010 and subsequently declined. Finally Fig. 2 presents a histogram of the out-degree distribution (aggregated over time) for directors, displaying a heavy right tail indicating that some directors have many links over the course of the study.

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Fig. 1.

Evolution of the proportion of interlocking links throughout the study. The interlocking proportion increased before and during the crisis, subsequently declining from 2010 to 2013.

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Fig. 2.

Histogram of out-degrees of directors aggregated over time indicating that some directors possess many connections throughout the time period analyzed.

A Dynamic Latent Space Model for Interlocking Boards.

The network is described by the adjacency cube Y whose elements are defined asyij(t)={1,if director i is a member of board j at time t,0,otherwise,

for t=1,…,T, i=1,…,N, j=1,…,M. To model the dynamic evolution of the bipartite network one could use a continuous-time framework (10). However, in the context of Irish interlocking directorates, we observe the board memberships on the last business day of each year, so each adjacency matrix represents a one-year aggregate snapshot of the boards and their directors. We use the term “aggregate” to describe the observed networks, but in reality, the boards of public limited companies are quite stable and it would be unusual for a director to serve less than one full term (a year). Thus, it is more natural to view the observations as equally spaced, and we make this assumption when formulating our model.

We use a latent space approach to model the positions of companies and their directors over time. This approach uses the idea that the positions of companies and their directors lie in a D-dimensional latent Euclidean space, so that a director which lies close to a company will tend to sit on that company board. We denote latent director and company positions at time t∈{1,…,T} by x1(t),…,xn(t) and w1(t),…,wm(t), respectively. An additional important aspect of our model is the inclusion of a edge persistence parameter, γ(t), which models the fact that once an edge is formed at time t, it may persist over time. We similarly include a nonedge persistence parameter, β(t) to capture the fact that a nonedge at time t also typically persists over time. Separating the edge and nonedge persistences is an important facet of our model, because the observed temporal bipartite graph is sparse and the effect of persistence of edges and nonedges is markedly different, as can be seen in Results and Discussion. Conditional on these latent positions, the log-odds of director i sitting on board j in year t islogit [pij(t)]=yij(t−1)γ(t)+[1−yij(t−1)]β(t)−‖xi−wj(t)‖.[1]

In fact, we found for the Irish boards data that modeling changes in director positions over time gave little extra information but was considerably more computationally demanding; hence, these are kept fixed. The persistence feature is included through the combination of the intercepts γ(t) and β(t) depending on whether an edge or a nonedge, respectively, is observed between nodes i and j at time t. If the state of the edge in the previous time frame is not known, the quantity yij(t−1) is substituted by a hyperparameter δ=0.0082, equal to the graph density. To summarize, the likelihood parameters are: the N×M×T observed adjacency cube Y, an N×D matrix X representing directors’ latent positions, an M×D×T cube W of boards’ positions and two T-dimensional vectors γ and β for the intercepts.

We account for the missing data by considering the contribution of boards only in those years where they are quoted in the ISE. As a matter of fact, from 2003 to 2013, no boards joined or left the ISE more than once. Hence, for each board the period of activity is made up of consecutive years only. A similar reasoning is applied to directors, whereas their contribution to the likelihood is considered only in the years where they sit on one board at least. The rationale leading to this simplification is that the likelihood information used to estimate positions is mainly carried by edges that are present rather than those that are missing. Hence, we speculate that neglecting the contribution of inactive boards and directors does not cause relevant changes in our conclusions.

Therefore, we express the likelihood of the network asℒY=∏t=1T∏j∈ℬt∏i∈Dtpij(t),[2]

where ℬt (Dt) denotes the set of active boards (directors, respectively) at time t.

Bayesian Model.

We formulate our dynamic latent space model as a Bayesian hierarchical model. We denote by f(s;μ,ν) a Gaussian density evaluated at s and with mean μ and variance ν, and by g(s;u,v) a gamma density evaluated at s with shape u and rate v. We use a random walk model for the boards’ latent positions, specified byπ(W|τw,τw0)=∏j=1M∏d=1Df(wjd(1);0,1τw0)×∏t=2T∏j=1M∏d=1Df(wjd(t);wjd(t−1),1τw).[3]

The directors’ positions are assumed to arise independently from a D-variate zero mean Gaussian prior with spherical covariance, henceπ(X)=∏i=1N∏d=1Df(xid;0,1τx).[4]

Note that the parameters corresponding to inactive boards and directors are included in the prior modeling. The intercept parameters follow a random walk prior:π(γ|τγ,τγ0)=f(γ(1);0,1τγ0)∏t=2Tf(γ(t);γ(t−1),1τγ);π(β|τβ,τβ0)=f(β(1);0,1τβ0)∏t=2Tf(β(t);β(t−1),1τβ).[5]

The precision parameters τw,τw0,τγ,τγ0,τβ,τβ0 are all modeled by gamma distributions with common shape a and rate b. (The precision is the reciprocal of the variance.) The remaining parameters τx, a, b are hyperparameters and hence user-specified. The hyperparameters were specified as τx=0.05, a=0.5, b=1 to reflect a relatively uninformative prior structure.

Markov Chain Monte Carlo (MCMC) sampling was used to sample from the joint posterior distribution of all of the model parameters. One issue is that the likelihood is invariant to rotations, reflections, and translations of the latent positions, because the likelihood depends on the latent positions through the distances ‖xi−wj(t)‖ for all i,j,t. To account for this source of model nonidentifiability, we apply Procrustes matching. Details of the MCMC algorithm and the Procrustes matching are given in the SI Appendix.

Quantifying the Extent of Interlockingness.

Quantifying the extent of interlockingness in a network of companies and directors may be of interest to policy makers and the public, especially in the aftermath of the economic crises in Ireland and beyond. A useful byproduct of the output of our statistical model is that the distribution of latent positions can be used to derive a metric of interlockingness in each year. In fact, we expect that directors located near the center of the space will more likely sit on more than one board, whereas those having a peripheral position will tend to have fewer connections. Equivalently, the centrality of boards affects the interlocking level of the network in the same way. For this reason, we assess the variation of interlocking through the empirical variance of boards, which may be computed for each iteration of the MCMC algorithm at each time t=1,…,T, asVt=1|ℬt|D∑j∈ℬt(wj(t)−w¯(t))′ (wj(t)−w¯(t)),[6]

where w¯(t)=∑j∈ℬtwj(t)/|ℬt| and dash refers to matrix transposition.

Model Choice.

We also compare our model with two simpler models for the same data. The three models are as follows:

• Model 1: the model described in this section.

• Model 2: a special case of Model 1, where the latent distances are all set to zero. Eq. 1 therefore simplifies to

logit[pij(t)]=yij(t−1)γ(t)+[1−yij(t−1)]β(t).[7]

• Model 3: a special case of Model 1, where the persistence feature is omitted. Therefore, the log-odds of Eq. 1 is expressed as

logit[pij(t)]=β(t)−‖xi−wj(t)‖.[8]

Model choice is carried out using the deviance information criteria (DIC) (25), which is estimated from the output of the MCMC algorithm.