Local Structure Graph Models, Parameter Centering & Higher-Order Dependence

Abstract:

This talk describes a new class of models for random graphs or network data called Local Structure Graph Models (LSGMs). LSGMs provide a Markov Random Field (MRF) modeling approach for random graphs, whereby each edge in the graph has a specified conditional distribution, i.e., probability of edge occurrence, dependent on explicit neighborhoods of other graph edges that impact a conditional distribution. As a consequence of the conditional specification, LSGMs have the advantage of allowing direct control and separate interpretation of parameters influencing large-scale (e.g., marginal means) and small-scale (i.e., dependence) structures in a graph model. This is possible through so-called centered parameterizations of MRF models, which are applied in LSGMs. However, current technology for centered parameterizations in MRFs assumes pairwise-only dependence, meaning that dependence is modeled between pairs of random variables only. This creates limitations in specifying conditional distributions for graph edges in LSGMs. As a remedy, we extend the centered parameterization for MRFs to account for triples of dependent edges in LSGMs. We also explain and numerically illustrate the importance of centered parameterizations when interpreting model parameters and, using a MRF framework, we additionally show that common exponential random graph models induce conditional distributions without centered parameterizations and thereby have undesirable consequences in parameter interpretation compared to LSGMs. Center parameterizations and their increased interpretation are particularly crucial when attribute/covariate information is included in a graph model. We illustrate these aspects for LSGMs with two network data examples.