Models for Network Evolution

This paper describes mathematical models for network evolution when ties (edges) are directed and the node set is xed. Each of these models implies a speci c type of departure from the standard null binomial model. We provide statistical tests that, in keeping with these models, are sensitive to particular types of departures from the null. Each model (and associated test) discussed follows directly from one or more socio-cognitive theories about how individuals alter the colleagues with whom they are likely to interact. The models include triad completion models, degree variance models, polarization and balkanization models, the Holland-Leinhardt models, metric models, and the constructural model. We nd that many of these models, in their basic form, tend asymptotically towards an equilibrium distribution centered at the completely connected network (i.e., all individuals are equally likely to interact with all other individuals); a fact that can inhibit the development of satisfactory tests.