Cycle Census Statistics for Exponential Random Graph Models

Exponential family models for random graphs (ERGs, also known as p ∗ models) are an increasingly popular tool for the analysis of social networks. ERGs allow for the parameterization of complex dependence among edges within a likelihood-based framework, and are often used to model local influences on global structure. This paper introduces a family of cycle statistics, which allow for the modeling of long-range dependence within ERGs. These statistics are shown to arise from a family of partial conditional dependence assumptions based on an extended form of reciprocity, here called reciprocal path dependence. Algorithms for computing cycle statistic changescores and the cycle census are provided, as are analytical expressions for the first and approximate second moments of the cycle census under a Bernoulli null model. An illustrative application of ERG modeling using cycle statistics is also provided