***** To join INSNA, visit http://www.insna.org ***** Guy Hagen wrote: > However, I am thinking it might be useful to be able to describe global > measures for such things. Perhaps, cliquishness= the percentage of > network actors that belong to identified cliquesets? global > equivalency = the percentage of network actors that belong to > identified structural/regular/eigenvector equivalent sets? > > I'll be interested in everybody's thoughts and/or citations. There are a number of tools in the literature which can be deployed in service of these questions. With respect to the former, the simplest measure is likely the density of triangles (i.e., the fraction of triads which are complete); the mean of the co-clique membership matrix (the adjacency matrix whose i,j entry is the number of cliques to which both i and j belong) is another. One can come up with numerous variants of these sorts of measures, and the best one(s) to deploy will obviously depend on the specifics of the problem under study. With respect to the second question, what you call the "equivalency" of a network is closely related to the algorithmic complexity of the associated graph. Intuitively, graphs which can be reduced to small blockmodels can be regenerated using a relatively small amount of information, and are hence "simple." One stab at this idea can be found in a paper of mine (see below), which applied a Lempel-Ziv based index to network data from a number of sources. Everett's role complexity index is another variation on this theme, as it effectively measures the extent to which positions within a graph are distinct under permutation. (Specifically, the role complexity index is 1-|Aut G|/|Perm G|, where Aut G and Perm G are the automorphism and permutation groups on G, respectively.) This, in turn, is closely related to Mowshowitz's orbit information index, which is the entropy of a random draw from the set of orbits (automorphic equivalence classes) on G. A discussion/comparison of these and several other measures can be found in the JMS paper listed below. Butts, C.T. 2000. "An Axiomatic Approach to Network Complexity." Journal of Mathematical Sociology, 24(4), 273-301. Butts, C.T. 2001. "The Complexity of Social Networks: Theoretical and Empirical Findings." Social Networks, 23(1), 31-71. Everett, M.C. 1985. "Role Similarity and Complexity in Social Networks." Social Networks, 7, 353-359. Mowshowitz, A. 1968. "Entropy and the Complexity of Graphs I: An Index of the Relative Complexity of a Graph." Bulletin of Mathematical Biophysics, 30, 175-204. Hope that helps, -Carter _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message.