Abstract
The most promising class of statistical models for expressing structural properties of social networks is the class of Exponential Random Graph Models (ERGMs), also known as $p^*$ models. The strong point of these models is that they can represent structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, MCMC algorithms have been developed which produce approximate Maximum Likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems and a poor fit to empirical data.
This paper proposes new specifications of the exponential random graph model. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistic are proposed: geometrically weighted degree distributions, alternating ktriangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.