Markov chain Monte Carlo exact inference for analysing social networks data

A test of reciprocity is often performed by analysts of social network data. This test corresponds to testing whether a parameter in an exponential family model for the adjacency matrix is zero. The uniformly most powerful unbiased test compares the observed number of mutual relations in the social network to its exact conditional distribution. As this distribution is typically only known to a constant of proportionality, Metropolis-Hastings algorithms have been proposed for generating from this distribution in order to perform Monte Carlo exact inference. Statistics based on the triad census are often used to test for the presence of group structure in a network. We show how one of the proposed Metropolis-Hastings algorithms can be modified to generate from the conditional distribution of the triad census given the in-degrees, the out-degrees and the number of mutual dyads. We compare the results of this algorithm with those obtained by using various approximations.