Structural equivalence of individuals in social networks

The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.

Notes

This work was supported under grants GS‐448 and GS‐2689 to H. C. White from the National Science Foundation, which is gratefully acknowledged. Special thanks are due to Scott Boorman, whose unfailing criticism gave us much cause for reflection and led us to numerous revisions. Discussions, in seminars or otherwise, particularly with Daniel Bertaux, Mark Granovetter, and John MacDougall, have also been helpful. However, the undersigned are clearly responsible for any imperfections that may remain. This paper is the product of the synthesis of an unpublished paper by H. C. White ('Notes on Finding Models of Structural Equivalence’, 1969) and certain results taken from an unpublished paper by F. Lorrain (Tools for the Formal Study of Networks, I.’, 1968) and from the latter's doctoral dissertation (Lorrain 1970, in press). The determinant stimulus that led us to the ideas set forth in the present paper came from Boyd's 1966 dissertation (the core of which was subsequently published—Boyd 1969), where the decisive step was made, introducing and exemplifying a particular type of reduction of a network. Friedell's (1967) notion of ‘office Structures’ within organizations also pointed in the same direction.