Enumeration and Simulation Methods for 0–1 Matrices with Given Marginals

Abstract

Data in the form of zero-one matrices where conditioning on the marginals is relevant arise in diverse fields such as social networks and ecology; directed graphs constitute an important special case. An algorithm is given for the complete enumeration of the family of all zero-one matrices with given marginals and with a prespecified set of cells with structural zero entries. Complete enumeration is computationally feasible only for relatively small matrices. Therefore, a more useable Monte Carlo simulation method for the uniform distribution over this family is given, based on unequal probability sampling and ratio estimation. This method is applied to testing reciprocity of choices in social networks.