Structural consequences of individual position in nondyadic social networks

JOURNAL

OF MATHEMATICAL

Structural

PSYCHOLOGY

29, 367-386 (1985)

Consequences of Individual Nondyadic Social Networks

Position

in

STEPHEN B. SEIDMAN Department

of Computer and Information George Mason University

Sciences,

The pattern of dyadic relationships among members of a population has often been used to obtain important information on the way that social structure can further or constrain social action. More recently, it has been proposed that nondyadic relationships can play a similar role. In this paper, a formalism is proposed that allows the dyadic structure and the nondyadic structure to be considered simultaneously. If this formalism is applied to the study of the structure arising from attendance at significant social events, it becomes possible to identify potential loci for social action. Still further, it is shown that information on individuals’ structural perspectives with respect to such potential loci can be translated into conclusions about the individuals’ positions in the dyadic structure. Such considerations can be used to evaluate the relative potential of these loci for social action. 0 1985 Academic Press, Inc.

1. INTRODUCTION It is generally accepted in social sciences that individuals’ actions can be supported or constrained by aspects of the social structure within which the individuals are situated. One particularly interesting aspect of social structure that has received much attention in recent years is the pattern generated by socially significant dyadic relationships, such as friendship, kinship, communication or aid. Such patterns of relationships have usually been called social networks. Anthropologists and sociologists have used a wide variety of structural features of social networks to demonstrate how social structure can support or constrain social action (see Holland & Leinhardt, 1979, for an overview). Important facets of social structure can often arise from nondyadic relationships as well (see Foster & Seidman, 1984; Seidman, 1981). Most commonly, nondyadic relationships are obtained from subsets of the population that can be given social significance. For example, the subsets can represent ceremonial events such as funerals and weddings. Since the subsets were chosen for their social significance, it is reasonable to claim that comembership in a subset may correspond to a significant tie between individuals. The research on which this paper is based was performed while the author was a Visiting Scholar in the Department of Anthropology at Arizona State University. It was partially supported by the National Science Foundation, Grant BNS 80-13507. The author thanks Brian L. Foster, William Batchelder, and Eugene Johnsen for many helpful discussions. Send requests for reprints to Stephen B. Seidman, Department of Computer and Information Sciences, George Mason University, Fairfax, VA 22030.

367 0022-2496185 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Structure is sought in the relationships between the pattern of comembership ties and the pattern of subset overlap. The analysis of social networks arising from dyadic relationships such as comembership ties has usually involved the construction of a graph, whose points correspond to the members of the population, and where two points are joined by a (possibly directed) edge if the corresponding individuals stand in the desired relationship to each other. Mathematically specified properties of such graphs can be used to define structural features of social networks that can be associated with sociologically interesting phenomena. The analysis of the pattern of subset overlap therefore requires the introduction of a mathematical object representing a collection of subsets of a set. Such an object is usually called a hypergruph (Berge, 1976). Just as a graph consists of a set of points and a set of (possibly ordered) pairs of those points corresponding to edges, a hypergraph consists of a set of points and a collection of subsets of that set. We usually denote a hypergraph as X = (X, a), where the hypergraph 2 consists of the set X of points and a collection 8 of nonempty subsets of X. By analogy with graphs, the members of X are called the points of the hypergraph and the members of 8 are called the edges of the hypergraph. In the situation described above, X is the set of individuals in a population, while 6 is the set of significant ceremonial events taking place, where each event is represented by the set of individuals that were present. Just as each event is a set of individuals, each individual can be regarded as the set of events that he or she attended. More generally, a hypergraph point can be made to correspond to the set of hypergraph edges that contain it. In this way, another hypergraph can be constructed, which is called the dual hypergruph of the original hypergraph. If the original hypergraph is 2’ = (X, E), the dual hypergraph of 2 is denoted X* = (8, X), since the points of YE’* are the edges of 2 and the edges of Xx* correspond to the points of 2”. For notational convenience, a point x E X (i.e., an individual) is denoted 3 if the point is regarded as a set of edges in 6. The hypergraph 2 = (A’, &) can also be used to define a graph G(8)). The points of this graph are the members of the set X, and two points x, , x2 of X are joined by an edge in G(X) precisely when the corresponding sets of hypergraph edges X, , 3, are not disjoint. Thus if X is the set of individuals in a population and d is a set of significant events, the individuals x1 and x2 are adjacent in G(X) precisely when the sets of events %i and X2 attended by x, and x2 have at least one event in common. Since this happens if and only if there is an event E E 8 attended by both x, and x2, it is natural to call the graph G(H) the graph of couttendance ties (or, more simply, the couttendunce graph). Note that G(X) is an undirected graph, since the coattendance relation is symmetric. More generally, for a positive integer q, we can define the q-ouerlup graph G,(X) to be a graph with point set X, for which points x,, x2 of X are adjacent precisely when X, and Z2 meet in at least q events. From this perspective, the coattendance graph is identical with the l-overlap graph. Figure la shows a hypergraph A?; the graphs G,(&) and G*(X) are shown in Figs. lb and lc. Since the dual hypergraph X* has too many edges to be easily depicted, Table 1 gives the points and edges of both 2 and Y?*.

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e.

FIG.

.f

1. The hypergraph shown in (a) yields the q-overlap graphs shown in (b) and (c).

Before we can ask how the patterns of coattendance ties and subset overlap interact, we must ask whether there are any particular types of structural features in either of these patterns that can be given theoretical significance. Since dyadic social networks have been the subject of intensive study for some time, it is easier to begin with the coattendance graph. Suppose, for example, that we intend to use the coattendance graph to investigate the social structure in a peasant village. If the social events that are used are chosen carefully, mutual attendance at an event creates a significant social tie between individuals. This tie may very well be weaker than a friendship tie, but it is likely to carry postive affect. Sociologists have long argued that members of “cohesive” sets of individuals in friendship networks tend to behave differently than nonmembers (see, for example, Coleman, 1960), even though the concept of “cohesive subset” was often left vague. Using the formalism of graph theory, it has been possible to outline more clearly what properties a “cohesive” subset of a friendship network should have. Seidman and Foster (1978b)

TABLE

1

A Hypergraph and Its Dual Hypergraph 2 = (X, 8) X={n,b.c,d,e,f} E, = {a, b, c) & = {c. 4 e, f)

d={E,,E,,E,,&} &= {b,e,f} Ed = {a, c, d} The dual hypergraph Z* = (8, X)

&= {E,, E,, Ex, &} a= j-E,, Ed} b={E,,E,j c= {E,, 6, &}

T={a,b,c,d,e,f} d= {&, E,} e= {El, E,} f= I&. 41

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have described three aspects of “cohesiveness.” First, members of a cohesive subset must be able to reach one another by a short path (reachability). Second, members of a cohesive subset must be joined by more than one short path (redundancy). Finally, a cohesive subset must be relatively invulnerable to the removal of a member (robustness), so that subsets of cohesive subsets also tend to be cohesive. This suggests that it is only necessary to look for maximal cohesive subsets of a population. These aspects are recognizable in all formal definitions of cohesive subsets in networks. The first such formal definition is due to Lute and Perry (1949), who proposed that maximal complete subgraphs (cliques) be used as cohesive subsets (see the Appendix for definitions from graph theory). Cliques clearly satisfy all three of the cohesiveness aspects outlined above, but they have not proved to be useful in analyses of most empirically occurring friendship networks, primarily because such networks have few nontrivial cliques. Several generalizations of the Lute and Perry clique definition have been proposed (Alba, 1973; Lute, 1950; Mokken, 1979), but the only proposal to specifically take account of all three aspects of cohesiveness is due to Seidman and Foster (1978a, 1978b). They propose the use of maximal kplexes (for small values of k) as cohesive subsets, and show that such k-plexes are strongly cohesive on all aspects. k-plexes have been used successfully in empirical analyses (see Foster, 1980). Since the coattendance tie carries positive affect, it is reasonable to argue that cohesive subsets of the coattendance graph are worthy of close attention. It is easy to see that each event gives rise to a complete subgraph of the coattendance graph, but such complete subgraphs need not be maximal (see Appendix). Thus there may exist cliques (i.e., maximal complete subgraphs) that do not arise from events. An example of such a clique can be seen in Fig. 1. In particular, the set {b, c, e, f > corresponds to a complete subgraph of G,(X), but {b, c, e, f} is contained in no single event of YE’. Cohesive subsets (cliques, k-plexes, or other structures) can therefore cut across several events. Suppose now that the overlap structure is generated by social events in a Thai village (Foster & Seidman, 1984). In this situation, dyadic relations such as kinship and friendship have not been found to be particularly useful in the study of social structure. Foster and Seidman argue that the overlap structure generated by social events forms the core of the village social structure (Foster & Seidman, 1984). If the goal were to investigate the occurence of political activity in such a village, it would be reasonable to hypothesize that cohesive subsets of the coattendance graph that cut across more than one social event are likely locations for such political activity (see also Foster & Seidman, 1982, 1984). We use the term pseudoevent to refer generally to a cohesive subset that is contained in no single event. When it is necessary to specify the type of cohesive subset that was used to define the pseudoevent, we avoid ambiguity by using the term “clique pseudoevent” or “k-plex pseudoevent.” We have already observed that {b, c, e, f> .is a clique pseudoevent in the hypergraph X depicted in Fig. 1. Similarly, the set {b, c, d, e, f} is a 2-plex pseudoevent in Z. Pseudoevents are thus found by identifying certain structural features of the coat-

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tendance graph that are contained in no single event. Although this already requires consideration of both the pattern of coattendance ties and the collection of subsets, we are able to investigate a far subtler aspect of the interaction of the coattendance graph and the subset overlap pattern. To see how this can be done, observe that all pseudoevents do not necessarily have the same potential for social action. One way of assessing this potential is to ask how the pseudoevents are situated in the subset overlap structure. For example, social action is more likely to occur in a particular pseudoevent if that pseudoevent contains an individual who can “see” the cohesion of the subset. Several natural definitions of just what it means for an individual to “see” cohesion (or other structural features) of a coattendance graph can be given, with rather different implications for the way in which the coattendance and overlap structures interact. In this paper, we discuss some of these definitions and their implications for the structure of nondyadic social networks.

2. COVERING NUMBERS OF SETS

Before introducing the various ways in which individuals can “see” coattendance network structural features, it is useful to introduce formalism that makes it possible to treat all of these ways from a common perspective. We have already seen that a hypergraph point can be associated with the set of edges that contain it; similarly, an edge in the coattendance graph can be associated with the set of hypergraph edges that contain both of its endpoints. Formally, we define duality functions d,,: X+28

byd,,(x)=

{E~dlx&)

and dedge:

E(G,(JU)

+2’

by

de,,,bY)

=

{EC

8 1 (4

Y>

=

El.

We refer to the set of points X and the set of (coattendance graph) edges E(G,(&‘)) as object sets 0, and introduce a general duality function d: 0 + 2”, where d(t) is the set of events of 8 that contain the “object” t. In a context implying 0 =X, d is taken to be dpt, while if 0 = E(G,(%)), d is taken to be dedge.By abuse of language, we also refer to subsets of object sets as object sets. From this perspective, the dual of a hypergraph point or coattendance graph edge corresponds to the events that contain that point or edge. Suppose now that we consider a set of hypergraph points or coattendance graph edges. It is natural to try to extend the duality concept to such sets. This can be done in two fundamentally different ways. First, the union of the duals of the elements of the set can be formed, to obtain the set of events containing any member of the object set. Using this idea, we can formally specify what it means for UN of the social activity of the

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members of an object set to be “visible.” In particular, that a set E c X is a wide cover of A if

if A is an object set, we say

(1) The second approach investigates the existence of a point dual that meets all of the (nonempty) duals of the elements of the set. We can now formally specify whether some social activity of each of the elements of the object set is “visible.” To make this precise, suppose that A is an object set. We say that a set E c X is a strict cover of A if for every a E A with d(u) # @, there is an x E E such that d,,(x) n d(a) +

0.

(2)

Note that if a is a coattendance graph edge, d(u) is necessarily nonempty; if a is a point, d(u) can only be empty if a is an isolate in the coattendance graph. It is clear that a superset of a strict or wide cover of A is also a strict or wide cover of A, and that a wide cover is always also a strict cover. On the other hand, we can ask how few individuals are needed to “see” the social activity of the members of an object set (from either perspective). We define the wide (strict) covering number of an object set to be the minimum of the cardinalities of the wide (strict) covers of A. In general, the strict covering number of an object set will always be less than or equal to the wide covering number. If a set consisting of a single point (x> is a wide (strict) cover of A (i.e., covering number = l), we also say that (the individual) x is a wide (strict) cover of A. If the object set of A is a set of points (i.e., individuals), we say that E is a wide point-cover of A or a strict point-cover of A if (1) or (2) holds, respectively. In this case, we also refer to the wide (strict) pointcovering number of A. If E = {x >, we say that (the individual) x is a wide pointcover or a strict point-cover of A (note that the covering number is 1). In this situation, the individual x can be said to “see” the social activity of the individuals who are members of A. For an example, consider the hypergraph J$? of Fig. 1 and Table 1. The set A = { 6, c, e, f} is a clique pseudoevent in X, and c is a strict point-cover of A. No single point is a wide point-cover of A, but the set {b, c} is such a wide point-cover. It follows that the strict point-covering number of A is 1, while the wide point-covering number of A is 2. Now suppose that A is the set of members of a cohesive subset of the coattendance graph that does not arise from a single event (i.e., A is a pseudoevent). We have argued that such a set A is a plausible locus for political activity growing out of the dense, crosscutting pattern of coattendance ties. The members of a (strict or wide) cover of A are collectively in a position to “see” the attendance patterns of the members of A; more concretely, if they shared their knowledge, they could deduce those attendance patterns. They are therefore well situated to initiate and direct political activity in A. On the other hand, large covers most likely find it difficult to initiate social action based on the structural knowledge of their members,

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especially in the absence of formal organizational structure. Thus the size of covers can be used to assess the degree to which a pseudoevent may be subject to political activity. The situation most conducive to political activity would therefore seem to occur when a pseudoevent has a wide or strict point-cover consisting of a single point x. The existence of a wide or strict point-cover E for a cohesive subset A of the coattendance graph clearly has structural implications, although these are not easily investigated if the size of E is arbitrary. The argument given above suggests that we should restrict our attention to relatively small covers, and that covers consisting of a single point should be of special interest. In this latter case, structural implications can be obtained fairly easily. For example, suppose that x is a strict point-cover of a pseudoevent S. It follows that for any point y of S, there must exist an event E with {x, y } c E. This means that x is adjacent to all points of S in the coattendance graph. Under any definition of cohesiveness that incorporates robustness and maximality, a point adjacent to all points of a cohesive subset is also a member of that subset. We henceforth require that a strict point-cover of a pseudoevent be a member of the pseudoevent. Such a pseudoevent must be contained in the star of the strict point-cover (see the Appendix for definitions). We also require that cohesive subsets have no isolates, so that strict covers have elements in common with all element duals. Suppose now that x is a wide point-cover of a pseudoevent S. In this case, x attended every event attended by any member of S. (Just as with strict point-covers, we require that x E S.) Individual x is thus potentially “aware” of all of the links into the larger population provided by the events attended by members of S. In other words, x can “see” the situation of the pseudoevent in the overlap structure. Such a wide point-cover would be in an excellent position to initiate social action within the pseudoevent in order to achieve results in the larger social field. As might be expected from their name, point-covers “see” structure as mediated by points; a strict point-cover (x ] of a pseudoevent E “sees” all of E, while a wide point-cover {x} of E “sees” the links to the complement of E that are provided by individuals’ attendance patterns. If we now set the object set 0 equal to E(G,(&?)), we can study the degree to which points can “see” the cohesiveness (or other properties) that arise from the incidence pattern among the edges of a pseudoevent. Suppose that S is a subset of X. Form the subgraph Gs of G,(X) induced by S, and let A be the set of edges of Gs. We say that a set E is a wide edge-cover of S or a strict edge-cover of S if (1) or (2) holds, respectively. The wide (strict) edge-covering number of S is defined to be the minimum of the cardinalities of the wide (strict) edge-covers of S. If {x} is a wide (strict) edge-cover of S, then we say that the point x is a wide (strict) edge-cover of S. If x is a strict edge-cover of S, then for every edge yz of G,, {x, y, z} is contained in some edge of 8. Thus x “sees” every edge of G,. By constrast, if x is a wide edge-cover of S, then for every edge yz of G,, x attends every event that is attended jointly by y and z. Thus, while a strict edgecover may “see” fewer links to the larger social field than a wide edge-cover, the links that it “sees” are sufficient to generate all of the pairwise coattendance links

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among the members of the cohesive subset S. For an example, consider once again the hypergraph &’ of Fig. 1 and Table 1. The set B = {a, b, c, d} is a 2-plex pseudoevent in 2, and c is both a strict and a wide edge-cover of 23.Thus the strict and wide edge-covering numbers of B are equal to 2. Less restrictively, we may only want to require that a point “see” various structural features of G,. For example, if Gs is connected, x will “see” the connectedness if it “sees” a collection of edges which together make up a connected subgraph of G,s. Similarly, if G, is a k-plex, x “sees” the k-plexness of G, if it “sees” edges that together make up a k-plex subgraph of Gs. Note, however, that the cliqueness of G, can only be “seen” by “seeing” all edges of Gs. To make this discussion formal, suppose that P is a structural property of a graph, whose presence depends only on the graph’s edge structure. Suppose further that the subgraph induced by the cohesive subset S has property P. We say that x is a wide (strict) P-cover of S if (1) or (2) holds, respectively, where A is a subset of E(G,) (incident with all points in S) that forms a subgraph of Gs that has property P (if A = E(G,), x is a wide (strict) edge-cover of S). For example, we may speak of wide clique-covers or strict 2-plex covers. Just as for point- and edge-covers, we can define the wide (strict) Pcovering number of S to be the minimum of the cardinalities of the wide (strict) Pcovers of S, and if {x} is a wide (strict) P-cover of S, we say that the point x is a wide (strict) P-cover of S. Note that P-covers and P-covering numbers are only defined for sets whose induced subgraphs have the property P. For an example, consider once more the hypergraph 2 of Fig. 1 and Table 1, with A = {b, c, e, f} and B = {a, b, c, d}. Then c is a strict (and wide) 2-plex cover of B, while no single point is a strict or wide 2-plex cover of A. Thus the strict and wide 2-plex covering numbers of B are both equal to 1. Since (b, c) is a strict (and wide) 2-plex cover of A, the corresponding covering numbers of B are equal to 2. The six types of covering numbers that we have discussed are clearly not independent. In particular, we have observed that the strict covering number of an object set is less than or equal to its wide covering number. This result can be applied to covering numbers that refer to points, edges, or structures. Furthermore, it is easy to see that if the subgraph G, has the structural property P, then the edgecovering number of S is greater or equal to the P-covering number of S, which in turn is greater or equal to the point-covering number of S. These inequalities hold equally for wide and strict covering numbers. Similarly, a (wide/strict) edge-cover of S must be a (wide/strict) P-cover of S, and a (wide/strict) P-cover of S must be a (wide/strict) point-cover of S. 3. STRUCTURAL FEATURES OF THE GRAPHS {G,J So far, our discission of overlap structure has focused on the identification of cohesive subsets of the population X, which have been identified as cohesive subgraphs of G,(X), and on the use of covers and covering numbers of various types to describe the ways in which the cohesive subgraphs fit into the overlapping sub-

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sets that form the edges of the hypergraph X. An alternative strategy for the analysis of the pattern of overlap is to investigate the structure of the graphs {G,(X)} for q = 1,2,.... Many investigators have used the connected components of these graphs as indicators of structure (see Atkin, 1974; Johnson, 1981; Seidman, 1981, 1983), and other graph-theoretic properties have been brought forth as candidates for use in future studies (Earl & Johnson, 1981). In this section, we use the concepts developed above to draw structural conclusions about the graphs (G,). In particular, we show that the existence of covers of various types of cohesive subgraphs of G,(X) has definite implications for the structure of G*(X). Suppose first that yz is an edge of Gk(Z) for some positive integer k, so that { y, z} is contained in at least k members of b. If 2 arises from individuals and the events they attended, y and z attended at least k events together. The intensity of the tie between y and z increases with k, but other individuals may “see” the intensity to different degrees. The degree to which an individual x “sees” the intensity of the tie between y and z can be measured by counting the number of events that contain {x, y, z}. More generally, if T is an arbitrary subset of X, we define the visibility of yz from T to be the number of events (hypergraph edges) that contain { y, z} and meet T. We denote this number by VT(yz). If T= {x}, V,(yz) counts the number of events that contain {x, y, z}. It is easy to see that 0 6 VAyz) < K, where K is the largest integer k for which yz is an edge in Gk(Z’). If we are investigating the structure of G,(s) and we observe that VT(yz) = k, we say that -7 is visibly adjacent to y from T, that T can see the edge yz, or that yz is visible from T. If the context permits, explicit mention of k is omitted. Using this terminology, and setting k = 1, we can restate some of our earlier definitions. If S is a subset of X, we see that a set T is a strict edge-cover of S precisely when each edge of the graph G, is visible from T. For a given property P (of graphs with vertex set S), T is a strict P-cover of S precisely when the edges of G, that are visible from T contain a subgraph of G, with property P. If property P is such that whenever G and G’ are graphs such that G has property P, V(G) = V(G’) and E(G’) 1E(G), it follows that G’ also has property P, we can replace “contain” by “form” in the preceding definition. Most properties that we deal with are of this type. For example, T is a strict 2-plex cover of S precisely when the edges of Gs that are visible from Tform a 2-plex with vertex set S (since adding edges to a k-plex only further strengthens its cohesiveness). While it is natural to expect that the existence of strict edge-covers will have implications for the structure of {Gk(Z’)}, it is more surprising that the same is true for strict P-covers, for many properties P. For a simple example, suppose that a subset S of the population induces a connected subgraph of G,(Z). That is, any two members of S can be joined by a chain of coattendance links. Suppose now that this connectedness can be “seen” (collectively) by the members of a subset T, in the sense that T is a strict connectednesscover of S. We can conclude (perhaps not surprisingly) that all members of S are within at most one coattendance link of T. To make this more formal, we have the following proposition (for definitions of graph-theoretic terms, see the Appendix).

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PROPOSITION 3.1. If T is a strict connectedness-cover for S c X, where ISI > 1, then the closed neighborhood N[ T] (in G,(X)) contains S.

Proof. By assumption, G, is connected. If S c T, the result follows trivially. Suppose that z E S - T. Since T is a strict connectedness-cover for S, the edges of G, that are visible from T form a connected subgraph of G, with vertex set S. This connected subgraph can contain no isolates, so that some edge yz must be visible from T. It follows that there is an event that contains {y, z} and meets T, say in a point x. We conclude that xz E G,( 2 )), as desired. 1 The strongest form of this result is obtained when T consists of a single point. In that case, we see that if a point x is a strict connectedness-cover of S (where ISI > 1) then x is adjacent to all (other) points of S. Far more interestingly, the existence of a strict P-cover for a property P may lead to structural conclusions about Gk(X) for k > 1. For example, we have: PROPOSITION 3.2. If x is a strict completeness-cover of S c X, where S is a nontrivial pseudoevent, then x is adjacent in G2(X) to all other points of S.

ProojI Let y #I be a point of S. By hypothesis, ISI > 1 and G, is complete, so that S must contain a point z # y and the edge y-7 is in G,. But since x is a strict completeness-cover of S, x can see the edge yz, so that there is an edge E E 47 containing {x, y, z}. Since S is a pseudoevent, E # S; choose w E S - E. But w must also be visibly adjacent to y, so that we must have {x, y, w} c E’, where E’ # E is an edge of 6. We conclude that {x, y ). c En E’, so that x is adjacent to y in GAW. I This result cannot be generalized to strict completeness-covers T with I TI > 1. To see why this is so, consider the hypergraph 2 = (X, &) with X= {x, ,..., x,} and ffT= u-5 XJi=2,....w (x,, x3,..., x,) ). In this hypergraph, the entire set X is a clique pseudoevent, and it is easy to see that although no single point is a strict completeness-cover, any two-point set of the form {x,, xi} “sees” the completeness of G,(X). Since the graph G*(X) is clearly totally disconnected, it follows that no conclusions as to the structure of GZ(X) can be drawn from the existence of strict completeness-covers involving more than one point. Thus the existence of a one-point strict completeness-cover for a pseudoevent has two important consequences. First, we can conclude that the graph G,(X) has some nontrivial structure (i.e., it is not totally disconnected). Second, such a cover is in a structurally powerful position; it is linked to all other points of the pseudoevent by strong ties (coattendance at more than one event). If cohesive pseudoevents are regarded as seedbeds of political activity, then it would seem that a clique pseudoevent with a one-point strict completeness-cover would be an exceptional candidate for such activity, since the cover would be linked to all other members of the pseudoevent by strong (and thus easily activated) ties. In hypergraphs obtained from sets of significant social events, clique pseudoevents generally arise from almost totally overlapping edges, and as a con-

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sequence are not much larger than the largest of those edges. In order to investigate the potential for political action of pseudoevents that crosscut social events to a larger degree, it would be interesting if Proposition 3.2 could be generalized to apply to less stringently constrained strict covers of less cohesive pseudoevents. A natural conjecture would be that if x is a strict k-plex cover of a pseudoevent S with ISI = n > 1, then x is adjacent to at least n-k points in G2(&‘). Note that Proposition 3.2 is obtained as a special case of this conjecture by setting k = 1. Unfortunately, a counterexample to the conjecture is provided by the hypergraph ~6 = (X, c?), where X= (a, b, c, d} and Q = ( {a, b, d}, {a, c, d] >. This hypergraph is shown in Fig. 2a; the graphs G,(X) and G,(Z) are shown in Figs. 2b and c. If we set S= X, both a and d are strict 2-plex covers of S, which is a 2-plex pseudoevent, but it is not true that the degrees of a and A in G2(%‘) are at least 2 = 4 - 2. The difficulty with this example is that the set S has more than one strict 2-plex cover. Surprisingly, if we require that the strict k-plex cover of S be unique, the conjecture is true. 3.3 Suppose that k 3 2 and n > 2k. Zf x is the unique strict k-plex cover of S c X, where S is a pseudoevent with n points, then deg,?(,* ,(x) > n - k. THEOREM

Proof Note that since S is a k-plex pseudoevent, G, is a k-plex with n points. If A is a subset of X, we write A= X-A. It suffices to show that if {y,,..., yk} are distinct points of S- {X ), then x must be adjacent in G?(H) to at least one of the {Yi}. Put Y={v I,..., yk}. Since S is connected in G,( 2) (Seidman & Foster, 1978a, Theorem 2) x must be adjacent in G,(X) to each point of Y. Without loss of generality, we can assume that all hypergraph edges that produce adjacencies in S are subsets of S. There must exist edges (E;},= ,,,,.,, such that {x) u Y c lJ:=, Ei, with YnE,#@ for each i, and XE~:=, E,. Suppose first that S# U:=, E,. In this case, there must exist a point ZES- lJisi E, that is visibly adjacent to at least one of the points of Y, say y,. But

.C

FIG. 2. The hypergraph shown in (a) yields the l-overlap graph shown in (b) and the 2-overlap graph shown in (c). The l-overlap graph has two strict 2-plex covers; the 2-overlap graph is disconnected.

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then {x, z, yi} is contained in an event different from all of the {Ei), and we conclude that xyj E G*(X), as desired. Assume now that S= Uf=, Ei. If t = 1, S= E,, contradicting the hypothesis that S is not an event. If a point y E Y is contained in two of the sets {Ei}, it follows immediately that xy E G,(X). Similarly, if a point y E Y n Ej is visibly adjacent to a point z E S- E,, it follows that xy E G,(X). We have thus reduced the proof to a consideration of the case in which (1) s= u:=, E;, (2) the events {Ei} partition the set Y, and (3) no point in Y n Ej is visibly adjacent j= 1, 2 )...) t.

to any point

of S- Ej, for

We now show that conditions (l), (2), and (3) imply the existence of a strict kplex cover z of S, where z #x. This contradicts the hypothesis that x is the unique strict k-plex cover of S, and this contradiction will complete the proof. Put m, = 1Yn Ejl; it is clear that mj 3 1 for j= 1, 2 ,..., t, and that cj=, mj = k. Choose a particular value ofj with 1 d j 6 t, and let y E Y n Ej. Since x is a strict kplex cover of S, at least n -k edges of S incident with y must be visible from x. Thus y can fail to be visibly adjacent to at most k-l other points of S. But, since we are assuming that (3) holds, y is not visibly adjacent to the xjzj mi = k - m,j points of Y-E!; we conclude that IS-E,-

YI dmj-

1.

(4)

Alternatively, y is visibly adjacent to x and to the other mj - 1 points of Y n E,. Since y is visibly adjacent to at least n - k points of S, we see that IE, - Y - {x} I > n -k - mj, and since n > 2k by hypothesis, we obtain IEj-

Y- {x)1 Zk-m,.

Note that the inequalities (4) and (5) hold for j= 1, 2,..., t. If Z is a subset of { 1, 2,..., t}, define

For example, Stii = Ejand Sfijj = (EinEj)-

(5)

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Let s, = (S,I and x1 = C,,, =, s,. Since all of the sets {S,) are disjoint tion, and since E, - Y - {x} = U iE, S,, inequality (5) yields

by construc-

for i= 1, 2 ,..., t.

1 s,>k-mi isl

Summing

over i, we obtain the inequality ...

x,+2x,+

2 (k-m,)

+(t-I)X,_,+tX,>

i=

1

or x*+2x,+

... +(t-1)x,-,+tx,>(t-l)k,

since 2 mi= k. i=

Similarly, inequality inequality

1

(4) yields C ,# I s, 6 mj - 1, and summing over j, we obtain the

(t-l)x,_,+(t-2)x,-,+

... +x,-,<

2 (mj-l)=k-t. j=

1

We have thus obtained the following inequalities: i i=

ixi>,(t-1)k

(6)

1

ig,(i-l)xi>t-k.

(7)

A k-plex pseudoevent satisfying the hypotheses and conditions (1 ), (2), and (3) thus gives rise to a simultaneous solution (x~}~= ,, 2,,..., of inequalities (6) and (7), where xi>0 for all i. But any such solution must also be a solution of any nonnegative linear combination of (6) and (7). In particular, it must be a solution of the inequality obtained by adding r - 1 copies of (7) to (6). But this inequality is i$, (i+(t-l)(i-t))xi>,(t-l)k+(t-l)(t-k), which can be rewritten as

$, t(i-(I-l))xi>t(t-11, or alternatively

as r-2

1 t(i-(t-l))xi>t(t-1)--x,. i=l

(8)

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Note that all the coefficients found on the left-hand side of inequality (8) are negative. Since xi > 0 for all i, we conclude that 0 2 t(t - 1) - txt, or that x, > t - 1. Since t > 2, it follows that x, > 0, which implies that n:= r Ei must contain a point z # x. We now complete the proof by showing that z must be a strict k-plex cover of S. We must show that z can see the fact that each point of S has degree at least n - k. First, let w be a point of n:= 1 Ei, and suppose that w’ E S is adjacent to w in G,(Z). Since w’ E some Ei by ( 1), {z, w, w’ ] E Ei, so that z sees the ww’ edge. Thus z can see all adjacencies involving w. Suppose now that w E S- of= r Ei, and assume that WEE,. Since by (4), IS-Ej-YI 2k requirement is met in Proposition 3.2 by the assumption that ISI > 1. We now present examples to show that the hypothesis of uniqueness and the requirement that n > 2k are both essential in Theorem 3.3. For the first example, suppose that n > 1 and m > 2 are integers. Construct a hypergraph Z,,” having edges A and B, where IA I = IB( = n + m and IA n BI = m. X= A u B, and 1x1 = 2n +m. The graph G,(Zn,,) is an (n+ l)-plex with 2n + m points, since the degree of each point is at least n + m - 1 = (2n + m) - (n + 1). Since m 2 2, the set X is a k-plex pseudoevent with at least 2k points, for any value of n. All m points in the intersection are strict (n + l)-plex covers of X. Thus, except for uniqueness, the hypotheses of Theorem 3.3 are satisfied. But if XE A n B, deg,,(,mnj(x) = m - 1 < (m - 1) + n = (2n + m) - (n - l), showing that the conclusion of the theorem does not hold if uniqueness is dropped from the hypotheses. Figure 2a shows the hypergraph x2,, , while Figs. 2b and c show the corresponding (graphs) GML 1 and G%,, 1. For the second example, let m > 2 be an integer. Construct a hypergraph &, having edges C, D, where (Cl = IDI = m and lCnDl= 1. Here, X=CuD and (X( = 2m - 1. The set X is an m-plex pseudoevent with 2m - 1 points, and the point x constituting Cn D is the unique strict m-plex cover. Note that the condition n >, 2k of Theorem 3.3 is not met. G,(xM) is totally disconnected, and in particular deg,,o,(x) = 0 < m - 1 = (2m - 1) - m, showing that the hypothesis n 2 2k cannot be removed from Theorem 3.3. Note that although the graph G,(xm) is connected, the point x is a cutpoint. On the other hand, it follows from (Seidman & Foster, 1978a, Corollary 5) that a k-plex with n points has no cutpoints if n > 2k. The converse of Theorem 3.3 does not hold, since points of high degree in subgraphs of G,(Y) need not be in a position to see cohesive structure in G,(Z). For

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an example, suppose that the hypergraph X = (X, b) has point set (a, b, c, d, e, f} and edge set b= {{a, b, c}, (b, c, d}, {a, b, e), {a, c,f}}. The graph G,(X) contains a 2-plex S induced by {a, b, c, d}. No single point is a strict 2-plex cover of S, even though deg @@f)(U)= 2 = 4 - 2. The structure of the set of strict covers clearly has a major role to play in the relationship between G,(Z) and G,(Z). We have seen that points that are strict completeness-covers or unique strict k-plex covers of pseudoevents in G,(X) must have relatively high degree in G2(#). The completeness result does not generalize to strict covers with more than one point. Any generalization of the k-plex result to larger strict covers requires a clear idea of what is meant by a “unique” strict cover. To make this precise, we define a minimal strict cover to be a strict cover containing no smaller strict cover. We can then conjecture that a unique minimal strict k-plex cover of a pseudoevent will have interesting structural properties in G2(X). This is certainly the case if the unique minimal strict cover consists of a single point. The following example shows that the conjecture is not likely to be true for larger unique minimal strict covers. Suppose that S is a k-plex pseudoevent in a hypergraph 2 = (X, &?), and also that x E X is a unique strict k-plex cover of S. Define a new hypergraph X” = (X’, &?‘), where x’ = Xu {z> (z $ X) and b’ = & u { { y, z> 1y E S}. If S’ = Su {z}, it is easy to see that Gs, is a k-plex in G,(#‘), so that S’ is a k-plex pseudoevent in 2’. Furthermore, since x is the unique strict k-plex cover of S, {x, 2) is the unique minimal strict k-plex cover of S’. Despite this, z is an isolate in the graph G,(Z’). Despite the fact that a minimal strict cover of a set S contains no smaller strict cover, it may contain a subset that is a strict cover of a subset T of S. In the preceding example, the point x was a strict k-plex cover of S c S’. To exclude this possibility, we say that a strict structure cover C of a set K is essential if for every x E C, C - {x} is not a strict structure cover of K- {x}. The following result shows that we can draw some (weak) conclusions from the existence of a two-point essential completeness-cover of a pseudoevent. PROPOSITION 3.4. If {x, y} is an essential strict completeness-cover of the pseudoeventK, where x E K and y E K, then deg.,(,,(x) + deg,,C,,(y) > 0.

Proof: If x and y are isolates in G,(P), then the edges of # that contain x and y, respectively, give rise to partitions P, and P.” of K,, = K- {x, y}. Suppose that S # K,, is a member of P,. Then every edge from a point of S to a point of K, - S must be visible from y. If S is contained in no member of P,, then there are distinct members T, T’ of P, that meet S. Choose points UE Tn S, u’ E T’n S, and u E KO - S. The edges uu and U’U must be visible from y, which implies that u must be in two elements of P,,, which is impossible. We conclude that if P, has any nontrivial members, those members must be subsets of members of P,. It is easy to conclude that we must have P, = P, = {K,), but this contradicts the assumption that {x, y} is an essential strict cover of K. 1

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The following example shows that this result is best possible. Define the hypergraph Z = (X, 8) to have point set X= {x, y, a,,..., a,>, where k> 2, and edge set E= {{x, y}, {x, uk}, {x, a,,..., u,-,>, {Y, u,, uk}, {Y, u2,..., %)I. It is easy to see that {x, y} is an essential strict cover of the clique G,(X), but deg,,&x) = 0 and dego,,.dy) = 1. These results and examples show that even if larger strict k-plex and completeness-covers are required to be minimal, unique, or essential, their existence does not lead to powerful structural consequences. It follows that if larger, unique minimal strict k-plex covers do have specifiable (and interesting) structural properties in G,, the circumstances will have to be defined far more precisely. The clearest link between the structure of G,(X) and that of G,(Z) is therefore provided by the degree to which there exist individuals who can “see” the cohesive structures in G,(X). For example, suppose that S is a subset of X for which G, is complete (in G,(X)). If there is an individual x E S who can see the completeness of Gs, then that individual will be adjacent to all other members of S in G,(X)). If there is no such individual, perhaps there is one who can see that G, is a k-plex, where ISI = n and k 6 4 n. If there is more than one such individual, we can draw no conclusions. If, however, there is precisely one, we can conclude that he/she will be adjacent to at least n -k other points of S in G2(%). If k > tn, we can again draw no conclusions. We can reason in exactly the same way if Gs is only assumed to be an I-plex, for 16 tn. The existence of a unique individual x who can see the I-plexness of Gs implies that x is adjacent to at least n - 1 other members of S in G2(X). If no such x exists and I < fn, perhaps we can find a unique x who can see the k-plexness of G,, where I< k < $PZ.Then x will be adjacent to at least n - k < n - I other members of S in G2(X)). 4. COMPUTATIONAL

CONSIDERATIONS

If the various types of covers introduced in this paper are to be used in empirical analyses that explore the hypotheses that have been proposed, it must be possible to find and compute them efficiently. The computation of wide covers requires the consideration of set inclusions such as

while the computation of strict covers requires the consideration of set intersections such as d,,(x)n d(u). Thus any algorithm for finding and computing covers and covering numbers must incorporate the ability to answer the fundamental settheoretic questions: (i) (ii)

for sets X, Y, is XC Y? for sets X, Y, is Xn Y # Qj?

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In most cases, the sizes of the sets X and Y under consideration are bounded by the number of points in a hypergraph or dual edge (and in the worst case by the number of points or edges in the hypergraph), and we assume that questions (i) and (ii) can each be answered in unit time. Similarly, we assume that the union of two sets can be performed in unit time, so that the time required to compute uaeA d(a) is no longer than (IAl - 1) times the time required to compute each dual. The time to compute a point dual should be roughly proportional to the size of the largest dual edge for the hypergraph, which we call R, and the time required to compute an edge dual is bounded above by 2R. Hence a time bound for the computation of UacA d(a) is either R(IAI - 1) or 2R(IAI - l), depending on whether point duals or edge duals are used. Since in most cases we only want to consider members of pseudoevents as potential covers of those pseudoevents, and since an inclusion can be tested in unit time, a rough upper bound for the time needed to search for a onepoint wide point- or edge-cover of a set A is tR IA I 2, where t is equal to 1 or 2. It is easy to see that the same time bound is valid for the determination of strict pointor edge-covers. Furthermore, the presence of unique point- or edge-covers can be determined within the same time bound. Complexity estimates for strict structure covers must be treated somewhat differently. We limit the discussion to complete subgraphs and k-plexes, all of which can be identified by examining degree sequences. Suppose that A is a clique or kplex pseudoevent. To determine whether a point x is a strict cover we must list the edges in GA that are visible from x. To do this, we can either look at each edge of G, and ask if it is visible from x or look at each edge visible from x and ask if it is in GA. The first approach requires mA operations, where mA is the number of edges in G,. Since A is a clique or k-plex, the order of magnitude of mA is [AI’. The second approach requires no more than 2R operations. Unless the hypergraph contains very many small edges, 2R is much smaller than IA12, and we assume that the second approach has been adopted. The edges obtained by this process are used to form a subgraph of G,, and this subgraph is then tested for the desired structural property. Forming this subgraph certainly needs no more than 2R operations. Since the test involves checking the degree of each vertex, it may require as many as IA/* operations. Finally, all this must be done for each x E A, so that an upper bound for the required time is given by 4R jA13, which is substantially larger than the bound for finding one-point wide covers. When we turn to the determination of larger covers of pseudoevents, we see that although the time needed for the computation is still polynomially bounded, the degree of the polynomial increases with the size of the cover. We can argue, however, that there is not much to be gained by trying to distinguish between pseudoevents with covering numbers 6 and 8, say; the only questions of empirical interest concern the existence or nonexistence of small covering sets (say with 1, 2, or 3 members). The determination of small covering sets has a time bound that is a polynomial of relatively small degree, but the computation is still likely to be infeasible if R or (A( is large. Such values of R or IAl are likely to occur if the hypergraph X (or its dual) consists of large highly overlapping subsets; in this case

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we are also likely to have many large pseudoevents in G,(X)), whose computation will also present problems. Empirically, covers and covering numbers are most likely to be useful in situations of moderate overlap among possibly large subsets. In such situations distinctions arising from covering numbers are likely to be both computable and salient. 5. CONCLUSIONS

An essential component of the study of structure arising from nondyadic relationships is the interplay between subset overlap and dyadic coattendance relations. The pseudoevent concept was introduced to identify cohesive subsets of the coattendance graph that did not arise directly from subsets. The various types of covers that have been defined in this paper distinguish points with respect to their positions in both the overlap structure and the coattendance structure. Although such distinctions between points are in themselves interesting and interpretable, we have shown that the existence of points in particularly strong positions in the l-overlap graph of a hypergraph has important implications for the structure of the corresponding 2-overlap graph. In particular, a point that can uniquely recognize the cohesiveness of a subset of the l-overlap graph must be tied to most members of that subset by double ties; in other words, the fact that a person is the only one who can see a cohesive structure defined by single coattendance ties implies that that person is in a privileged and very strong position with respect to double coattendance ties. This conclusion has clear implications for the definition of network positions that have leadership potential. The complexity of the algorithms needed to locate the covers defined above is highly dependent on the size and overlap pattern of the subsets used to form the hypergraph under study. It can be argued, however, that hypergraphs for which covers and covering numbers are essentially uncomputable reflect situations in which those covers and covering numbers are unlikely to have much social significance. For example, if a hypergraph consists of 50 points and 10 edges, where the edges each have between 25 and 35 points and overlap heavily, the overlap graphs will be exceptionally dense. If cliques and k-plexes are used to identify pseudoevents, there will be many large pseudoevents, each differing from the others in only a few points. In this situation, it will not be feasible to compute covers and covering numbers, but it seems clear that the events corresponding to the subsets overlap so much that it is hard to imagine that knowledge of the overlap pattern could be socially significant. On the other hand, if the hypergraph edges had roughly 10 points each and relatively small overlaps, it will be much easier both to identify pseudoevents and compute their covers and covering numbers. It will also be far more natural to believe that people in a position to have unique knowledge of subset cohesiveness will be able to make use of that knowledge in organizing their behavior. We can thus conclude that the conceptual framework provided by covering numbers and covers of various types serves two distinct, complementary purposes. First,

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it enables us to distinguish between empirical situations in which subset overlap structure is likely or unlikely to provide information useful for individuals’ social actions. Second, within situations for which subset structure can provide useful information, it enables us to distinguish between pseudoevents with greater or lesser social potential and to locate individuals who are in strong positions to utilize that potential in their own social action.

APPENDIX

All sets considered here are finite. The cardinaliry of a set A is the number of points in A, denoted IA I. A graph G consists of a set of points V(G) and a set of edges E(G), where each edge in E(G) joins a pair of points in V(G), and no pair of points in V(G) can be joined by more than one edge in E(G). Points x, y are said to be adjacent if they are joined by an edge. A path from a point x0 to a point x, is a sequence of alternating points and edges x0, e,, x1,..., e,, x,, where e, is an edge from xi-i to xi. If a path can be found between any two points of V(G), the graph G is connected. The star of a point x E V(G) is the set N(x) = { y I y is adjacent to x}. The degree of x is the number of points adjacent to x. It is denoted deg (x), so that we have deg (x) = IN(x The degree sequence of a graph is the sequence of degrees of the points of the graph, arranged so that they are in nonincreasing order. Generalizing the concept of star, we define the neighborhood of a set of points S c V(G) to be N(S) = { y $ S 1y is adjacent to some x E S}. The closed neighborhood of S is defined to be N[ S] = S u N(S). If S c V(G), the subgraph of G induced by S is the graph G, with I’( G,) = S and E(G,)={~EE(G)I e j oms ’ two points of S}. A graph is complete if any pair of its points is joined by an edge. If a complete graph has n points, each of those points has degree n - 1. A clique of a graph G is a maximal complete (induced) subgraph of G. That is, if a clique of G is induced by the set S, then the subgraph induced by S u {x} is not complete for any x $ S. For a positive integer k, a k-plex is a graph with n points for which each point has degree at least n -k. Note that a 1-plex is a complete graph. If k < (n + 2)/2, a k-plex with n points is connected (Seidman & Foster, 1978a). A hypergraph 2 consists of a set of points X and a set of edges b, where each edge in d is a nonempty subset of S. Note that a graph can be regarded as a hypergraph whose edges are two-point subsets of X. We write 2 = (X, 8). The dual hypergruph #* of the hypergraph Y? is #* = (8, X), where each point of X is regarded as the set of edges that contain it. For more information on the concepts defined above, see Berge (1976), Harary (1969), and Seidman & Foster (1978a).

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RECEIVED: February

14, 1984