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June 17, 2006 Network Position Measures Roles and positions are central concepts in social network analysis. Analyses of interlocking roles using algebraic and matrix methods began with S.F. Nadel’s (1957) theory of social structure as positions possessing distinctive rights and duties in relation to other positions. Structural equivalence is one fundamental method for identifying roles and positions in a social network. However, its requirement that structurally equivalent actors must have identical patterns of ties the same other actors is too stringent for much practical use. Although loosening the criterion to permit similarities broadens the method’s applicability, structural equivalence still imposes a very restrictive notion of which actors jointly occupy positions based on their structural relations. As an alternative, network researchers developed several less-restrictive approaches to identifying roles and positions (Everett 1985; Faust 1988; Pattison 1988; Everett, Boyd and Borgatti 1990; Borgatti and Everett 1992). In decreasing order of restrictiveness, structural equivalence is the most restrictive, followed by automorphic and isomorphic equivalence, then regular equivalence. This section provides a nontechnical overview of those measures. For simplicity of illustration, we examine a nondirected binary graph of a single relation, although with some modifications, both automorphic and isomorphic equivalence can be applied to directed and valued graphs (Wasserman and Faust 1994:461-502). Isomorphic and automorphic equivalence are such closely related concepts that some researchers treat them as interchangeable (Borgatti and Everett 1992). However, isomorphic equivalence applies to two graphs, whereas automorphic equivalence describes the relational properties of social actors within one graph. Two graphs exhibit structural isomorphism if a oneto-one mapping of the nodes from one graph onto the second graph preserves all nodes’ adjacency relations (i.e., the same indegrees and outdegrees). In other words, if two nodes are 1 June 17, 2006 connected in the first graph, then the corresponding two nodes in the second graph must also be connected in the same way (Borgatti and Everett 1992:11). Every graph is isomorphic with itself, which is called automorphism, a one-to-one mapping of nodes back onto themselves. Two actors are automorphically equivalent (jointly occupy the same position) if and only if they are connected to corresponding other positions (not to identical nodes). Automorphic equivalent nodes have identical graph theoretic properties, such as centrality, ego-density, and clique size (Borgatti and Everett 1992). Automorphic equivalence relaxes the structural equivalence requirement that requires actors in the same position have identical ties with the same set of other actors. Instead, automorphic equivalence identifies actors as jointly occupying a position if they have identical ties with different sets of actors that play the same role in relation to the position. To use a familiar example, for two professors to occupy a structurally equivalent position, both must teach the identical set of students, which is a virtual impossibility. But, to occupy an automorphically equivalent position, the two professors need only teach different sets with the same number of students. The students occupy a second position, defined as persons taught by the professor position. The graphs in Figure 4.10 contrast these two types of equivalence, where directed lines from professors to students represent the “teach” relation. Although the two graphs in have two positions, automorphic equivalence better captures the idea that social roles involve generalized patterns of relations. To cite another well-known instance, we expect the wife role to be jointly occupied by women who are not in plural marriages to the same set of men, but are each uniquely paired with a different husband! Structurally equivalent actors are also automorphically equivalent, but not necessarily vice versa. Automorphically equivalent nodes are indistinguishable if the actor labels are 2 June 17, 2006 removed from a graph. Thus, if points are substituted for the names in Figure 11B, the two subgraphs cannot be told apart. Borgatti and Everett (1992:16) summarized the distinction: Abstracting a bit, we could say that in the structural equivalence approach, the network or labeled graph represents the underlying structure of a group; hence an actor’s location in that structure represents his or her position in the group. In contrast, in the [automorphic equivalence] approach, the structure of interest is not the labeled graph itself, which is seen as the observed or “surface structure,” but the structure of the surface structure, which is the unlabeled graph that underlies the labeled graph. It is the actor’s location in this “deep structure,” then, that represents his or her position in the group. By relaxing the structural equivalence requirements, automorphic equivalence proves very useful in facilitating empirical research corresponding to many social theories. Borgatti and Everett (1992) summarized and clarified several studies using structural equivalence to operationalize different theories, which, in fact, are better operationalized via automorphic equivalence. For example, they addressed Ronald Burt’s (1979) proposal to define the industries and sectors of an economy as sets of firms that produce similar types of goods and occupy a single position within an interorganizational network. Borgatti and Everett (1992:21) argued that structurally equivalent firms, which by definition must buy from the same providers and sell to the same clients, hardly constitute recognizable sectors. But, automorphically equivalent firms, which buy from similar vendors and sell to similar customers, might well comprise meaningful industries and sectors. 3 June 17, 2006 Regular equivalence is the least restrictive of the three most commonly used forms of equivalence. It requires neither structural equivalence’s ties to identical actors nor automorphic equivalences’ indistinguishable positions. Actors are regularly equivalent if they have the same kinds of relations with actors that are also regularly equivalent. Another way to conceptualize the idea is that, if a first actor occupying a position is tied to someone in a second position, then a regularly equivalent second actor must have an identical tie to a someone else in that second position (White and Reitz 1983:214). All mothers with children are regularly equivalent, regardless of their numbers of offspring, as are all children who have mothers. In a hospital, the doctors are regularly equivalent in relation to their patients and nurses. The generality of regular equivalence makes it perhaps the most important measure for sociologists attempting to capture social roles and positions. The following paragraphs review studies on the definitions of equivalence (Borgatti and Everett 1992; Borgatti and Everett 1993; Everett 1985; Borgatti and Everett 1989; Doreian 1987; Everett, Boyd, and Borgatti 1990; Faust 1988), attempting to clarify the differences between the three types of equivalence. [These works, most cited in the first paragraph, are not reviewed below.] Both automorphic equivalence and regular equivalence require that a pair of actors connect with the other actors who are structurally equivalent on the same relation. However, the distinction between automorphic and regular equivalence is sometimes unclear. Automorphic equivalence requires that unlabelled graphs be substitutable for one another, but regular equivalence does not require a complete substitutability between subgraphs. To demonstrate the difference, Figure 4.12 depicts a hierarchical network of an imaginary organization consisting of four vertical levels linked by supervisory relations. The CEO supervises three executive managers (A, B, C), who supervise four middle managers (D, E, F, G), who in turn supervise a 4 June 17, 2006 few front-line employees (H through N). If we ignore the employees, then executive managers B and C are structurally equivalent because both have identical supervisory ties to the same middle managers (F and G). But, A is not structurally equivalent to B and C, because A supervises different middle managers. However, the three executives are regularly equivalent, since each supervises the same number of middle managers. If we consider all hierarchical levels, B and C are also automorphically equivalent because their subgraphs are substitutable for one another if the labels are removed. But, A’s subgraph cannot be substituted since its two middle managers supervise three front-line employees, while both B’s and C’s subgraphs have four employees. Considering only the two lowest levels, none of the four middle managers are structurally equivalent, because they all supervise different front-line employees. Instead, three of the middle managers ( D, F, and G) are automorphically equivalent, because their two-supervisee subgraphs are completely substitutable once the labels are removed (unlike E with only one supervisee). But, all four middle managers meet the regular equivalence criterion by supervising at least one employee. Figure 4.12 demonstrates that structural equivalence is the most restrictive form, regular equivalence is the least strict, and automorphic equivalence lies in between. Regular equivalence seems a very flexible method for identifying generalized social roles in networks, broadly defined as aggregate classes or categories of actors having similar structural relations with other positions in a social system (Faust 1988:315). 5