Discrete exponential family models for random graphs (ERGs) are increasingly popular tools for the analysis of discrete relational data. 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 presents 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. One important application of the above model family arises in the context of subjects' subjective evaluations of social interaction within their local environment (sometimes called "cognitive social structures"). Balance theory, in particular, posits a tendency towards dyadic and triadic closure in positive relations. Implications of balance theory for the properties of long-cycle structure have been derived by Harary and others, but empirical evaluation has been hindered by the lack of statistical models for cycles of length greater than 3. We here use Markov chain Monte Carlo methods to fit ERG models for biases in cycle structure formation to two sets of 21 cognitive social structures from managers in a high-tech manufacturing firm. Implications of the estimated structural parameters for modeling of cognitive social structures are discussed. |