structural equivalence: meaning and definition, computation and
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Social Networks, 1 (1978) 73-90 @Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
Structural Equivalence: Meaning and Definition, Computation and Application*
Lee Douglas Sailer
University of cizlifornia, Irvine**
This paper presents a generalization of the concept of Structural equi- valence * the key concept in algebraic approaches to the study of social networks. Two points in a graph or set of relations will be called Struc- turally related if they are connected in the same ways to structural1.y related points.
It is suggested that this new definition suitably weakens Lorrain and Whites cutegorical approach, and is more appropriate. than CONCOR. St~~t~~ra~ relatedness is compared to these approaches via several simple examples.
This paper is concerned with the algebraic approach to network analysis (though it isnt always called that) exemplified by White (19631, Boyd (1969), Lorrain and White (197 l), Boyd et al. (197 l), Breiger et al. (1975), Heil and White (1976), White et al. (1976), Boorman and White (1976), D. White and Boyd (1977), and Boyd and Sailer (1978). These algebraic ideas provide an underpinning for theories of social structure. These theories, in turn, may be used to derive computational definitions which accurately reflect interesting aspects of the system under investigation. It is possible to argue that the approach in this paper is consistent with the writings of Linton (1936), Nadel(1957), Merton (1957) and Goodenough (1969).
The concept of structural equivalence (sometimes called here SE) is examined in four stages. First, the relationship between structural equi- valence and standard sociological concepts is discussed. Second, various
*In no way could this paper have been written without the patience, persistance, and perspicacity of John Boyd and Douglas White. In addition, many of these ideas were a group creation of a seminar at UC Irvine in 1977. This research was supported in part by NSF grant #BNS 76-08386, D. R. White, H. Nutini, L. Brudner, Principal Investigators. **Division of Social Sciencks, University of California, Irvine, Calif. 92717, U.S.A.
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more detailed definitions are contrasted. Third, the details of one defini- tion and a computational procedure for realising it are described. Fourth, and finally, some applications are proposed.
Social structure, roles and status
Social structure is the network of actually existing social relations, or at least so said Radcliffe-Brown (1943: 190). Social structure may be taken to contrast with such systems as environment, language, and beliefs. Another common contrast is with function, content .and process. To under- stand my interpretation of the quote above a careful distinction must be made. A social relation is not the tie between specific people, but rather a set of ties in an entire population. The network is the patterning of social relations over a set of persons, or positions, or groups, or organizations. The individual is important primarily as the vehicle for the extensive definition of the relations in which we are interested. People are not social structure; the interaction of people may be structured, but here I am concerned with the interaction of the interactions. The interplay between patterns of kin ties and patterns of economic ties, for example, is more interesting than (or at least separate from) the relationship between two specific groups of people. From this view, the specific actors are transitory. They may change roles through time even though the social structure remains the same. (See Nadel 1957: 16 - 17 for an early statement similar to this.)
Goodenough (1969), following Linton, Merton, and Nadel, distinguishes carefully between status, the rights and obligations of a role, and social identity, for Goodenough the label for the special position occupied by one of the actors in a specific relationship. For Goodenough, the methodolo- gical task is to find the rules which will translate informants beliefs about the role structure (where role is the totality of status, social identity, and other constructs) into behavior.
The approach here is different, but congruent. From data on the occur- rence of interactions (or of beliefs about them), can we define models for which it will be easy to find rules relating the model to behavior? The social scientist has two tasks before him then, to categorize relations, and to describe the relational interactions. These tasks may entail the initial cate- gorization of the actors in the network, i.e., by function.
Most occurrences in this paper of the technical (i.e., mathematical) term relation may be loosely replaced by the term role. For the purposes of this paper, a role is a set of appropriate behaviors exhibited by a pair of actors in a particular context. A role may also be defined as a cover term or gloss for certain individual attributes (which may be network attributes),
The terminology used here is Goodenoughs. The tarn role is used loosely
Structural equivalence 75
that is, a role is a function fulfilled by an individual. Important to the con- cept of role is the notion that a particular individual has certain connections to other individuals, and that those others must themselves be in certain positions.
Here it is assumed that roles are patterns of ties in observable and un- observable relations. The importance of the reciprocal nature of roles is thus sidestepped. The relations represent the actual occurrence of behavior, and if subsets of the populations are found to be reciprocal (e.g., doctors and nurses), then let this be an external validation of the method. Certainly, if husband and wife are relations in a data set, we would expect to find that husbands have wives, but this is a different aspect of the problem. A thorough treatment of this topic is necessary.
What is structural equivalence?
One of the easiest things to see in a network is who is connected to whom. That is probably why so much effort has been expended in the development of cluster and clique detection techniques. Cliques, however, are not to be confused with roles. Here is an example of the difference. One would expect the cliques in a kin network to be entities such as families, clans, lineages, etc. The roles of interest in a kin network, though, are kin-types, e.g., father and son. Families are certainly interesting structures in their own right, but they are not roles. Rather, they are nodes in a higher order relation. Clusters and cliques, as structures, correspond to such concepts as the family or clan. The mathematical structure which corresponds to the role, such as father or boss, is the block. A blockmodel is a set of such blocks and the relation- ships between them.2 The cluster concept is still relevant, however. A block can be defined as a set of actors clustered together by virtue of their structural equivalence.
Blocks are not defined by the amount of intra-role interaction, as are clusters, but by the intrinsic nature of the other blocks with which they connect; e.g., judges interact with layers more than they do with other judges; crooks interact with victims in a different way than they do with other crooks. Notice that it is assumed that status, the rights and obliga- tions of a role, is contained in or expressed by the patterns of ties labeled by the name of the role. That is, it is exactly the actual exercise of rights and obligations (and/or the expectations of them) that we take as data, using these data to discover the roles themselves.
Two people in the same role are substitutable. This is what structural equivalence is, substitutability with regard to relational ties. Relational structure does not totally determine SE properties, though. Certainly, a great
*Blockmodel is used in a loose sense here, contrasted with clique structure for example. Heil and White (1976) provide an example of a very formal definition. The spirit is the same.
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deal of information can be obtained much information can we tease out of tion asked here.
Definitions of structural equivalence
from individual attributes, but how relationships alone? That is the ques-
Lorrain and White ( 197 1:63) label as structurally equivalent any two points that are related in the same ways to the same other points. That is, for i, i, and k in a set N, and relations (sets of ordered pairs) R,, R,, . . . . R, in N X N, a relation S is a structural equivalence if iSj implies that, for every k, iRk = jRk and kRi = kRj for each relation Ri.,
Of course, hardly anything or anybody is ever structurally equivalent to anybody else in the noisy and complex world of social relations. To over- come this obstacle. Lorrain and White apply what they call the categorical approach. They use various criteria to reduce the number of relations derivable from the data, thus aggregating relationship data to enable nodes to be blocked together.
Figure 1. Map of the algebraic approach to the analysis of social structure.
-\ approach '-._
-.fcf combine -9,) combine
CONCOR -+ Model B
Furthermore, they claim (p. 79) that this approach treats the blocking of the nodes and the aggregation of the relations simultaneously, as suggested by the analogue of Category Theory. This, in fact, is not true. They identify relations (morphisms in their terminology) first, with no reference to blocks at all, and only subsequently use this information to identify nodes. Later in this paper a technique for blocking