Abstract
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.
Highlights
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
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
There are some circumstances or types of roles in which it is expected that all occupants of one role will be identically connected to occupants of counterparts roles. This may be formally defined in terms of a structural homomorphism familiar to graph theory (Hedetniemi, 1966; Lorrain 1974; Arabie, Boorman and Levitt 1978), as follows
Summary
A function f: P + P’ is a mapping of each element a in the set P to an image element f(a) in the set P’. An equivalence = on P is a relation such that for all a, 6, c in P, a=a a = b implies b = a (a = b and b = c) implies a = c. These are the properties of reflexivity, symmetry, and transitivity. =! on P, namely, for each u, b E P, a=,bifandonlyiff(a)=f(b) The proof of this lemma is found in standard texts. R’ is called the relation on P’ induced by R and f
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