Estimation for Dyadic-Dependent Exponential Random Graph Models

Abstract

Graphs are the primary mathematical representation for networks, with nodes or verticescorresponding to units (e.g., individuals) and edges corresponding to relationships. ExponentialRandom Graph Models (ERGMs) are widely used for describing network data because of theirsimple structure as an exponential function of a sum of parameters multiplied by their correspondingsucient statistics. As with other exponential family settings the key computational dicultyis determining the normalizing constant for the likelihood function, a quantity that depends onlyon the data. In ERGMs for network data, the normalizing constant in the model often makesthe parameter estimation intractable for large graphs, when the model involves dependence amongdyads in the graph. One way to deal with this problem is to approximate the likelihood functionby something tractable, e.g., by using the method of pseudo-likelihood estimation suggested inthe early literature. In this paper, we describe the family of ERGMs and explain the increasingcomplexity that arises from imposing dierent edge dependence and homogeneous parameter assumptions.We then compare maximum likelihood (ML) and maximum pseudo-likelihood (MPL)estimation schemes with respect to existence and related degeneracy properties for ERGMs involvingdependencies among dyads.