ERGM with unknown normalizing constant

Abstract:

Exponential families of probability measures has been a subject of considerable interest in Statistics, both in theoretical and applied areas. This talk will mostly concentrate on Exponential Random Graph Models (in short ERGM), i.e. exponential families on the space of graphs. One of the problems that frequently arise in such models is that the normalizing constant is not known in closed form. Also numerical computation of the normalizing constant is infeasible because the size of the space becomes large, even when we consider moderate sized graphs. As such, carrying out inferential procedures such as MLE is hard to do. Also, it has been empirically observed in social science literature that choice of some sufficient statistics for the ERGMs makes the model “degen-erate”, thus making such models unsuitable for modeling purposes.
This talk will address these issues, and gives a framework for analyzing a class of sparse ERGMs using Large Deviations Theory. The large deviation used is for the empirical degree distribution of an Erdo˝s-Renyi graph, with a topology stronger than weak convergence. This analysis gives an approximation for the normalizing constant when the size of the graph is large. It also gives one quantification for degeneracy, and gives a simple check to see if a particular ERGM is non- degenerate. As an application, we explain non-degeneracy of certain graph statistics popular in the social science literature which are known to not cause degeneracy at an empirical level.