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 k-
triangles, 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 in-
crease the range and applicability of the ERGM as a tool for the statistical analysis of social
networks.
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