The question invariably arises as to whether graphs can accommodate social relations that differ in strength, number, or type. By suitably modifying a graph or digraph to permit the assignment of values to its lines or arcs, the answer is in the affirmative for many, although obviously not for all, kinds of empirical structures. Fig. 7.1, for example, depicts a social network, adapted by Patrick Doreian (1974), from Bruce Kapferer's (1969) analysis of a conflict in a Zambian work unit. The points represent men of the work unit, and the lines, social relations between them. The relations include (1) conversational exchange, (2) joking exchange, (3) job assistance, and (4) personal service. Any single relation is designated by Kapferer as “uniplex,” and any multiple relation as “multiplex,” represented by the thin and thick lines, respectively. The possible values of the lines are 1 or 2. (The values could also be 1, 2, 3, 4 if we chose not to lump multiplex relations.) The interest of this diagram, to which we will return, centers on the strength, as inferred from the multiplexity, of the direct and indirect links mobilized in support of two disputants.

The assignment of values to the lines, together with the relaxation of the stricture on loops, results in a very general model, of which a social network is only one interpretation. The values can represent such things as flows, probabilities, sequences, costs, and strengths in a variety of networks. We will offer three interpretations.