HOLLAND, P., LASKEY, K.B., and LEINHARDT, S. (1983), "Stochastic blockmodels: Some first steps", Social Networks, 5, 109 -- 137.  Home/Search   Document Not in Database   Summary   Related Articles

....of this equivalence concept was discussed by White, Boorman and Breiger (1976) and this concept is now established as a major tool in positional role analysis of social networks. Extensive discussions are given in Faust (1988) and Wasserman and Faust (1994) Fienberg and Wasserman (1981) and Holland, Laskey, and Leinhardt (1983) generalized the deterministic concept of structural equivalence to probabilistic models. They introduced probability models for directed graphs (where y ij can have values 0 and 1) for which the distribution of the relation y ij depends on actor attributes, under the additional assumptions of ....

....where e a (k, h) a # A, 1 # k # h # c, are the relation counts for block (k, h) defined by e a (k, h) 1 I k = h I a # A 0 ) 1 (8) # (i,j)#N I y ij = a I x i = k I x j = h , and I A is defined as 1 if condition A is satisfied, and 0 otherwise. This gives an extension of Holland, Laskey, and Leinhardt s (1983) concept of a pair dependent stochastic blockmodel ; in their paper the studied structure was restricted to digraphs, i.e. # = 0, 1 and A = 0, 1 2 . The stochastic block model is then given by the joint distribution of (Y, X) P(y, x #, #) # m1 1 . # m c c (9) # # # ....

HOLLAND, P., LASKEY, K.B., and LEINHARDT, S. (1983), "Stochastic blockmodels: Some first steps", Social Networks, 5, 109 -- 137.

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