A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others wHolland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33–65 Ž with discussion.; Fienberg, S.E., Wasserman,

Abstract: 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. Keywords: triad completion, Holland-Leinhardt model, polarization, degree variance, network evolution, constructuralism

*We acknowledge the helpful comments of the editor and anonymous reviewers. For their encouragement and suggestions on the research, we thank H. Russell Bernard, Linton Freeman, and A. Kimball Romney. We thank Tracy Burkett and Douglas Nigh for making their data available to us.Comparing Networks Across Space and Time, Size and Species We describe and illustrate methodology for comparing networks from diverse settings. Our empirical base consists of 42 networks from four kinds of species (humans, non-human primates, non-primate mammals, and birds) and covering distinct types of relations such as influence, grooming, and agonistic encounters. The general problem is to determine whether networks are similarly structured despite their surface differences. The methodology we propose is generally applicable to the characterization and comparison of network-level social structures across multiple settings, such as different organizations, communities, or social groups, and to the examination of sources of variability in network structure. We first fit a p * model (Wasserman and Pattison 1996) to each network to obtain estimates for effects of six structural properties on the probability of the graph. Then we calculate predicted tie probabilities for each network, using both its own parameter estimates and the estimates from each other network in the collection. Comparison is based on the similarity between sets of predicted tie probabilities. We then use correspondence analysis to represent the similarities among all 42 networks and interpret the resulting configuration using information about the species and relations involved. Results show that similarities among the networks are due more to the kind of relation than to the kind of animal. 2