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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
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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
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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.
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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
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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).
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