Exponential random graph model specifications for bipartite networks—A dependence hierarchy

Abstract

In this paper, we review the development of dependence structures for exponential random graph models for bipartite networks, and propose a hierarchy of dependence structures within which different dependence assumptions may be located. Based on this hierarchy, we propose a new set of model specifications by including bipartite graph configurations involving more than four nodes. We discuss the theoretical significance of the various effects that the extended models afford, and illustrate application of this hierarchy of models to several bipartite networks related to the political mobilization in Brazil in the early 1990s (Mische, 2007).

Introduction

Exponential random graph models (ERGMs), also known as p* models, introduced by Frank and Strauss (1986) and Wasserman and Pattison (1996) allow global network structure to be understood by considering/focusing on local processes. Such models assign probabilities to networks based on local network configurations which representing the local network processes. The local network effects included in a particular model specification are based on dependence assumptions between network tie variables. From the Bernoulli random graph models (Erdös and Rényi, 1960) to the recent social circuit models (Snijders et al., 2006, Robins et al., 2007a, Robins et al., 2007b), the ERGM specifications for one-mode networks are enriched by various forms of dependence assumptions and the inferred endogenous network effects. The tie dependence assumptions not only generate the model specifications but also provide guidance on the interpretation of estimated effects.

Based on an analogous set of dependence assumptions, ERGM specifications for bipartite networks were developed by Skvoretz and Faust (1999), Pattison and Robins, 2002, Pattison and Robins, 2004, Agneessens and Roose (2008) and Wang et al. (2009). Bipartite ERGMs with exogenous effects were also proposed by Agneessens et al. (2004) and Agneessens and Roose (2008). MCMC maximum likelihood estimation methods (e.g. Snijders, 2002) provide us with more reliable model estimates than the older pseudolikelihood approaches (Frank and Strauss, 1986, Strauss and Ikeda, 1990; see also van Duijn et al., 2009, for a comparison), so that these models can be estimated in a principled way. Wang et al. (2009) introduced an ERGM specification for bipartite networks which adapted the social circuit model for one-mode networks (Snijders et al., 2006), and applied a geometric weighting on related graph statistics to alleviate the issue of model degeneracy (Handcock, 2003).

In this paper, we review and discuss the development of dependence structures for exponential random graph (p*) models for bipartite networks, and propose a hierarchy of dependence structures within which different dependence assumptions may be located. The motivation behind developing such hierarchy is that first of all, the current ERGM specifications have difficulty in model convergence for some observed networks; and, secondly, not all converged models provide good fit to the observed network. These two points suggest the need for more elaborate model specifications which may include more complicated graph statistics. This raises the question of how potential ERGM specifications may be developed in a systematic manner. The proposed hierarchy is based on graph theoretical properties, and aims to provide a theoretical framework to extend the current set of dependence hypotheses for ERGMs, such that new configurations can be introduced to the model specifications in a principled way. It is not intended to solve the model degeneracy problem with ERGMs, or to provide general rules for parameter selection in specific models. However, such issues may be alleviated by using the specifications we introduce based on the proposed hierarchy.

We generate the hierarchy by a generalized dependence assumption governed by two factors: the form of proximity between a potentially dependent pair of ties; and the graph theoretical distance with which that proximity is described. This hierarchy can be used to guide the exploration and development of models for observed bipartite network structures. As dependence becomes more complex within the hierarchy, more relaxed conditions govern the pairs of network ties that are considered conditionally dependent, and more complex network effects are introduced into ERGMs for bipartite networks. As a result, we propose a systematically elaborated set of model specifications which extend the models discussed by Wang et al. (2009) and include bipartite graph statistics involving more than four nodes. In particular, we focus on the new edge-cycle network configuration which represents an interaction between network closure and activity. Similar geometric weighting techniques are used for definitions of the higher order configurations to assist model convergence in MCMC estimation. We discuss the theoretical significance of the various effects that the extended models afford using simulations, and illustrate application of this hierarchy of models to bipartite networks related to the political mobilization in Brazil in the early 1990s (Mische, 2007). The simulation and modelling examples are conducted using the BPNet programme as an extension to PNet (Wang et al., 2006).