The algebraic definitions presented here are motivated by our search for an adequate formalization of the concepts of social roles as regularities in social network patterns. The theorems represent significant homomorphic reductions of social networks which are possible using these definitions to capture the role structure of a network. The concepts build directly on the pioneering work of S.F. Nadel (1957) and the pathbreaking approach to blockmodeling introduced by Lorrain and White (1971) and refined in subsequent years (White, Boorman and Breiger 1976;Boorman and White 1976; Arabie, Boorman and Levitt, 1978; Sailer, 1978). Blockmodeling is one of the predominant techniques for deriving structural models of social networks. When a network is represented by a directed multigraph, a blockmodel of the multigraph can be characterized as mapping points and edges onto their images in a reduced multigraph. The relations in a network or multigraph can also be composed to form a semigroup. In the first part of the paper we examine "graph" homomorphisms, or homomorphic mappings of the points or actors in a network. A family of basic concepts of role equivalence are introduced, and theorems presented to show the structure preserving properties of their various induced homomorphisms. This extends the "classic" approach to blockmodeling via the equivalence of positions. Lorrain and White (1971), Pattison (1980), Boyd (1980, 1982), and most recently Bonacich (1982) have explored the topic taken up in the second part of this paper, namely the homomorphic reduction of the semigroup of relations on a network, and the relation between semigroup and graph homomorphisms. Our approach allows us a significant beginning in reducing the complexity of a multigraph by collapsing relations which play a similar "role" in the network. © 1983.

A modification of Fisher's exact test for the 2 x 2 x 2 contingency table is proposed as a test of the null hypothesis of no three-way statistical in teraction among variables, controlling for the two-way or first-order correlations. The test uses a truncated hypergeometric distribution, limited by the bivariate marginal totals of the variables. Possible generalizations to L x M x N tables are discussed. The test is also applicable to the null hypothesis of no difference in the magnitude of correlation in a comparison of two bivariate distributions. Illustrations of each application are provided. One obvious use in cross-cultural or survey research is as a test of the replication of a correlation in different subsamples of a population. © 1983, Sage Publications. All rights reserved.

Classical statistical inference procedures usually assume the independence of sample units. However, the assumption of independence is often unrealistic in cross‐cultural research because societies in neighboring or historically related regions tend to be duplicates of one another across a wide variety of traits that are spread by historical fission, diffusion, or migration of peoples. A recent generalization of the usual regression model explicitly allows for networks of interdependencies among sample units as part of the model specification. Here, two new estimation procedures for this network autocorrelation model are compared to previously employed maximum likelihood procedures, and to the usual regression procedures which ignore interdependence. The results of comparisons based on simulated autocorrelation data and the reanalyses of two previously published empirical studies indicate that both of the procedures proposed here compare very favorably with the maximum likelihood approach, and both are vastly superior to the usual regression procedures when there is moderate to high autocorrelation (i.e., interdependence). [Galton's Problem, cultural diffusion, networks, cultural evolution, statistical methodology] 1984 American Anthropological Association