See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/233251812 A graph‐theoretic generalization of the clique concept* Article in Journal of Mathematical Sociology · January 1978 DOI: 10.1080/0022250X.1978.9989883 CITATIONS READS 217 224 2 authors, including: Stephen B. Seidman Texas State University 67 PUBLICATIONS 1,470 CITATIONS SEE PROFILE All content following this page was uploaded by Stephen B. Seidman on 13 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. J.Mathematical Sociology 1978, Vol. 6, pp. 139-154 ©1978 Gordon and Breach Science Publishers, Ltd. Printed in the United Kingdom A G R A P H - T H E O R E T I C G E N E R A L I Z A T I O N OF THE CLIQUE CONCEPT* Stephen B. Seidman Department of Mathematios, George Mason University and Brian L. Foster Department of Anthropology, State University of New York at Binghamton For at least twenty-five years, the concept of the clique has had a prominent place in sociometric and other kinds of sociological research. Recently, with theadvent of large, fast computers and with the growth of interest in graph-theoretic social network studies, research on the definition and investigation of the graph theoretic prop-erties of clique-like structures has grown. In the pres-ent paper, several of these formulations are examined, and their mathematical properties analyzed. A family of new clique-like structures is proposed which captures an aspect of cliques which is seldom treated in the exist-ing literature. The new structures, when used to comple-ment existing concepts, provide a new means of tapping several important properties of social networks. I. INTRODUCTION Many anthropologists and sociologists have sought to analyze social structure by somehow breaking up the social unit under analysis into subgroups and then examining inter- and intra-subgroup relations. Anthropologists have tended to concentrate on groupings defined normatively and embedded in complex kinship networks ( e . g . , lineages, kindreds, households). Perhaps the closest sociological analogues to anthropologists' kinship studies are found in research on formal organizations, where formal role structures provide an overarching relational reference point similar to the kinship networks of primitive *The network research on which this paper is based was carried out under support from the National Science Foundation, Grant No. BNS 76-05023. The Thai data were gathered by Foster in Thailand from August 1970 through December 1971; the field research was supported by the Foreign Area Fellowship Program of the Social Science Research council and the American Council of Learned Societies. 139 I 140 S. B. SEIDMAN AND B. L. FOSTER peoples. In general, however, sociologists lack such an overarching relational system, and they have taken a more empirical approach. Their social groupings generally have been either categorical (e.g., occupational or religious) or they have been derived from sociometric data or from techniques for measuring face-to-face interaction. Much sociological research of this kind has centered around the analysis of cliques, which almost universally has been concerned with sets of individuals who are tied to each other more closely than to non-members. Although this very general and informal notion is widely used, few investigators have attempted to formally define the concept. In addition, different researchers have focused on very different theoretical aspects of such structures and have devised quite different procedures for finding them in sociometric or other kinds of network data. The most widely used formal definition of a clique is that of Luce and Perry (1949) , in which a clique is a maximal complete subgraph of the graph representing the population under study. Use of this concept of clique in substantive analysis has, however, presented several difficulties. First, at the time the Luce and Perry article appeared, methods for finding such structures in even moderate sized networks were unfeasible, although the development of large computers has made this restriction no longer critical. Second, the definition is so restrictive that interesting clique structures are seldom found, except in networks that are so dense as to be of little structural interest. Third, since complete graphs contain paths of length one from each point to every other point, internal structure is of no interest; alternatively, defining cliques in this way suggests no natural means of investigating the internal structure of empirical examples, since by definition all possible links exist. In subsequent years, each of these difficulties was attacked in various ways. A large number of clique detection methods was devised, few of which were accompanied by formal definitions. Some early techniques focused on re-arranging sociomatrices so that the non-zero cells fell along the main diagonal, thus facilitating visual identification (e.g., Coleman & MacRae, 1960; Spilerman, 1966). Some methods were based on taking powers of matrices (e.g., Festinger, 1949; Hubbell, 1965) . Most methods, however, were in some sense statistical (see Lankford, 1974, for a review of several such methods). One interesting method combines statistical analysis with the investigation of paths of minimum length in the graph_(Killworth s Bernard, 1974). In addition, efficient algorithms were devised to search for maximal complete subgraphs in even large graphs (Johnston, 1976). At the same time, work proceeded on formal modifications of the clique concept in ways which would help with the second and third problems. Luce (1950) defined the n-clique, which "loosened" the clique structure to allow for indirect links of various lengths, some of which might even occur outside the clique itself. The n-clique notion was modified and developed further by Alba (1973), who also provided efficient search routines and placed these structures in a broad theoretical GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 141 and methodological framework. We will return to Alba's work shortly. A second graph theoretic approach, rather than redefining the graphic structure itself, developed methods for working with valued graphs and in so doing redefined the relations (e.g., Doreian, 1969; Peay, 1974). Although these two general approaches to the study of cliques are fundamentally different, they are by no means contradictory and could profitably be combined. In different ways each suggests important new directions in network studies. First, each allows cliques with varying degrees of strength to be defined, thus mitigating the excessive restrictiveness of the Luce and Perry formulation. Second, each provides a way of investigating important dimensions of internal structural differentiation. In Alba's formulation, complete graphs are weakened by looking at structures of different strength (e.g., at sets of individuals represented by graphs of diameter 1, 2, 3, etc.), while the valued graph approach consists of the definition of true cliques on relations of different strengths and, therefore, the definition of hierarchically ordered cliques within cliques (Doreian, 1969; Peay, 1974). Most important, however, by retaining the graph theoretic orientation, both of these approaches allow the clique structures to be placed in the global network in such a way that no information on either the clique's internal structure or its relationship to the rest of the network is lost. This raises a wide range of new questions concerning the trans-formation of cliques (e.g., by individual action) in view of social constraints and resources inherent in the structural properties of the total network. For example, it is possible to investigate the structural potential of clique-like structures for growth or for structural strengthening. Similarly, the vulnerability of such structures to random demographic events or to competitive recruitment by other cliques can be studied. Our own proposals are concerned with defining graph theoretic, clique-like objects of varying strength and with investigating the graph theoretic properties of the structures. Like Alba's approach, ours varies the strength of the structure rather than the relations, although it is in no way incompatible with valued graph approaches. Before turning to a technical discussion of the mathematical properties of the structures, which we call k-plexes, it is necessary to discuss briefly the theoretical motivation for defining the type of structure we propose. II. THEORETICAL MOTIVATION FOR THE K-PLEX CONCEPT In order to define clique-like structures of variable strength, it is necessary to vary in controlled degrees two internal properties, both of which are inherent in the property of completeness. The first is the degree to which a short path is present from each point to every other point in the subgraph. According to the most common theoretical interpretation of this property, its sociological importance stems from the direct communication ties among the individual members. For some purposes, when considering this reachability, S. B. SEIDMAN AND B. L. FOSTER 142 it is useful to distinguish between the directness of the ties and the degree to which the star (in Barnes's, 1972, sense) of any arbitrary point spans the structure. The second prop-erty, which follows directly from the first, is the robustness of the structure. This property, which is seldom discussed in literature on cliques, is best characterized with refer-ence to the degree to which the structure is vulnerable to the removal of any given individual. The degree of robustness of individual cliques, in turn, is also closely related to the stability of the clique structure of the total network. A third property which is sometimes considered in defining cliques is the degree to which such structures are tied into the total network. That is, it is sometimes said that a defining property of cliques is that they be disjoint and/or that they show marked discontinuities with the rest of the network (Alba, 1973: 121). While the degree of discontinuity with the total network is an important structural property, it is a property of a somewhat different order than the previous two-—i.e., it is a positional rather than an intrinsic property (see Poster & Seidman, n.d. (b), for an extended discussion of this distinction) . This property will not be discussed here except to note that the relative importance of intrinsic and positional properties for analysis of the clique structure of the global graph varies with the degree of completeness of the individual cliques; the less they are complete, the greater the attention which must be given to their internal properties in analyzing the global structure. The proposals in this paper are motivated, then, by a desire to vary simultaneously and in controlled degree the directness and the star-coverage aspects of the reachability property of clique-like structures, and it is in precisely this way that our proposals differ most significantly from Alba's formulation. Alba defines his diameter-n cliques as maximal subgraphs of diameter n. The "true clique" (i.e., complete subgraph) is thus a special case in his diameter—n clique scheme (i.e., a diameter-1 clique). Although the directness aspect of reachability is varied in controlled degree as n is changed, other critical structural properties disappear altogether even at the diameter-2 clique level. This is so, because, for example, any graph of the form K1, n (i.e., a graph in which one point is connected directly to all others and there are no other lines) is a diameter-2 clique. Such structures are very vulnerable, their robustness diminishing from maximal to possibly minimal levels in the transition from graphs of diameter 1 to those of diameter 2. Accordingly, the clique structure of the global graph may toe unstable due to the vulnerability of the constituent units. The k-plex structure which we propose here is defined as a graph with n vertices in which each vertex is connected by a path of length 1 to at least n- k of the other vertices. (A formal definition and a discussion of the mathematical properties of k-plexes is found below.) By controlling in this way the degree to which the star of any arbitrary point covers the graph, several other properties of true cliques are preserved. First, there is a close relationship between the degree of star coverage and reachability, as can be seen GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 143 by the fact that a k-plex with n points has diameter at most 2 if k <(n + 2)/2. Thus, although only the starcoverage aspect of the reachability property is explicitly controlled, the directness aspect follows implicitly. Second, the structures are robust in the sense that if as many as n - 2k + 1 arbitrary points are removed, the resulting graph will have diameter of at most two. Third, and closely related to the level of robustness, any subgraph of a k-plex is a k-plex. Fourth, true cliques fit naturally into the k-plex pattern (k = 1) , just as they fit into Alba's diameter-n clique scheme . In addition to sharing these properties with true cliques , k-plexes have an important characteristic that facilitates investigation of the ways in which they can be transformed into other graphic structures, the ways in which other graphic structures can be transformed into kplexes, or ways in which the combination of other structures produces k-plexes. In particular, where k is small, the internal structure of k-plexes is highly determined by the values of k and n and by degree sequences. By contrast, for diameter-n cliques, variation of internal structure is much greater, and in fact, internal structure even of diameter 2 graphs is poorly understood (Harary & Palmer, 1973: 223). The utility of the Luce and Perry clique concept and of all the other methods discussed above stems from their formal mathematical status. This is equally, true of the k-plex scheme, which has not only been formalized in and of itself, but which has also been placed in an explicit relationship to true cliques and to the structures defined by Alba. It is, therefore, necessary to turn now to a detailed discussion of the mathematical properties of kplexes. III. MATHEMATICAL PROPERTIES OP K-PLEXES Basic Definitions Let G be a graph with n vertices, and let V(G) denote its vertex set. If v ∈ V ( G ) , S(v) will denote the set consisting of v and all vertices adjacent to v, µ(G) will denote the minimum degree of the vertices of G, δ(G) will denote the diameter of G, and κ( G ) will denote the connectivity of G. κl will denote the complete graph on l vertices. The degree sequence of G is the sequence {deg(v)} v ∈ V(G). If S is a finite set, then |S| will denote the cardinality of S. Definition and Basic Properties of K-plexes A graph G will be called a k-plex for some positive integer k if µ(G) ≥ n - k. Since complete graphs are 1plexes (and indeed k-plexes for any k), the concept of kplex generalizes that of complete graph. An alternative characterization of k-plexes is given by the following result: Theorem 1: G is a k-plex if and only if for any subset 144 S. B. SEIDMAN AND B. L. FOSTER k {v1, ..., vk} consisting of k vertices of G, V(G) = ∪ S(vi). i=l Proof: Note that the case for k = 1 is obvious. If G is a kplex, deg(v) ≥ n - k for each v ∈ V(G) . Let {v1, ..., vk} be a subset of V(G). For each i = 1, ..., k, put Li = V(G) - S(vi). Observe that ∩ Li ≠ ∅, since if v ∈ ∩ Li, k k i=l i=1 v is not adjacent to v, v1, ..., vk, so that deg(v) < n - k, which is impossible. Thus ∅ = ∩ Li = ∩ (V(G) – S ( v i ) ) = k k i=1 k i=1 k i=l i=l V(G) - ∪ S(Vi), so that V(G) = ∪ S ( V t ) , as desired. Conversely, if G is not a k-plex, there exists v ∈ V(G) not adjacent to vertices V1, ..., vk (v ≠ vi). But then clearly v ∉ ∪ S(vi), so that V(G) ≠ ∪ S(vi). K k k i=l i=l Thus, a graph is a k-plex precisely when any k stars cover the graph. This property of k-plexes holds independently of the size of the graph. More detailed information about k-plexes can be obtained by requiring k to be small relative to n. In particular, we have the following results: ≤ 2. Theorem 2: If G is a k-plex and k < (n + 2)/2, then δ(G) Proof: Suppose that v ∈ V(G); since G is a k-plex it follows that |s(v)| ≥ n - k + 1. Thus if u, v ∈ V(G), |S (u)| + |s(v)| ≥ 2n - 2k + 2 > n by hypothesis. Hence the sets S(u) and S(v) are not disjoint, which implies that d(u,v) ≥ 2. Thus δ(G) ≤ 2, as desired. The following result is an immediate consequence of the alternative characterization of k-plexes given in Theorem 1: Proposition 3: Any subgraph of a k-plex is a k-plex. Thus if we remove & points from a k-plex G, we obtain a k-plex G1 with n - I points. By Theorem 2, 6(G') <. 2 if k < (n - I + 2)/2, so that we obtain the following generalization of Theorem 2: Theorem 4: If G is a k-plex and l < n - 2k + 2, then if G' is a graph obtained from G by removing I vertices, δ(G') ≤ 2. An immediate consequence of Theorem 4 is the following result of Chartrand and Harary (1968): Corollary 5: If G is a k-plex, then κ(G) ≥ n - 2k + 2. GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 145 The following examples show that the bounds given in Theorem 2 and 4 cannot be improved: (a) (b) Let Gl consist of two copies of Kl+1 joined by one line. Gl has n = 2l + 2 points, and µ(Gl) = l. Thus Gl, is a k-plex with k = l + 2 = (n + 2)/2 points, but δ(Gl) = 3. Let Hn,k = H1 ∪ H2 ∪ H3, when H1 = Kn-2k+2 and H2 = H3 = Kk-l where k > 1 and n - 2k + 2 > 0, and the only lines not contained in the Hi_join all points of H2 to all points of H1 and all points of H3 to all points of H1. Hn,k has n points and µ(Hn,k) = n - k. Thus Hn,k is a k-plex with n points, but if the n - 2k + 2 points of H1 are removed, the resulting graph is disconnected. Structure of 2-Plexes In order to obtain detailed information both about the number of k-plexes with n points and about the internal structure of those k-plexes, we must restrict our attention to small values of k. We will begin with k = 2, since the only k-plexes for k = 1 are the complete graphs. Theorem 6: A 2-plex is determined uniquely by its degree sequence. Proof: Let G be a 2-plex and let H = Kn - G be its complement. Then the degree sequence of H is (1, ..., 1, 0, ..., 0) 2b n – 2b for some integer b. But the only graph corresponding to this degree sequence is the disjoint union of b copies of K2 and n - 2b copies of K1. Since H is uniquely determined by its degree sequence (and hence by the degree sequence of G), so is G. It follows that the number of 2-plexes with n points is [n/2], where [x] denotes the largest integer ≤ x. (Note that if we want to include Kn as a 2-plex the number should be [n/2] + 1.) The complement can also be used to investigate the internal structure of 2-plexes. An independent set of vertices of G is a subset of V(G) such that no two of its vertices are joined by a line in G. Let G be a 2-plex, and as in the proof of Theorem 6, let H be its complement, where H is assumed to be the disjoint union of b copies of K2 and n - 2b copies of K1. In particular, we put V(H) = b ∪ {xi,Yi} ∪ {zi,..., zn-2b}. For each subset A ⊂ {1,2,..., b}, there is a corresponding maximal independent subset of H 146 S. B. SEIDMAN AND B. L. FOSTER {xi}i∈A ∪{yi}i∉A ∪ {z1, ..., zn-2b}, and all maximal independent subsets of H can be obtained in this way. Thus the maximal independent subsets of H are in one-to-one correspondence with the subsets of {1,2, ..., b}; it follows that there are 2b such maximal independent sets, each with n - b points. But the subgraphs of G induced by the maximal independent sets of H are precisely the cliques (the maximal complete subgraphs) of G, so that we have shown that G has 2b cliques, each with n - b points. The clique graph of a graph G is a graph with points representing the cliques of G, in which the points representing two cliques are adjacent if the cliques have nonempty intersection. If 2b < n, each pair of cliques of G intersects, so that the clique graph of G is complete on 2b vertices. If 2b = n, let S = {1,2, ..., b>. All cliques of G are of the form KA, where V(KA) = {xi }i∈A ∪ {yi }i∉A. But then KA ∪ KB = ∅ if and only if A ∩ B = ∅ and (S - A) ∩ (S - B) = ∅. Thus KA ∩ KB = ∅ if and only if B = S - A, and we see that the clique graph of G is regular of degree 2b - 2. Note that we have shown that the clique graph of a 2-plex is itself a 2-plex. Structure of k-plexes for k > 2 If k > 2, k-plexes are no longer determined uniquely by their degree sequences, as is illustrated by the two 3-plexes in Figure 1, each with 5 points and degree sequence (3,3,2,2,2). Figure 1 Two 3-plexes with identical degree sequences. Although degree sequences for the two graphs are identical, the graphs are not isomorphic. Some results can be obtained for k = 3, although they are not as complete and satisfying as the results presented above for k = 2. We consider a 3-plex G with n points, and put H = Kn - G. The degree sequence for H must be of the form (2,2,...,2 1,...,1 0,...,0), where a and b are intea 2b n-a-2b gers. All such sequences are graphical except those corresponding to a = 2, b = 0 and a = 1, b = 0. Thus we can enumerate the number of possible graphical complementary degree sequences for given n. There are n - 1 such sequences for GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 147 which b = 0, while if i is given (1 ≤ I [n/2]), there are i such sequences for which a = n - 2i and b > 0. Hence the total number of graphical complementary degree sequences is Thus we have determined the number of degree sequences that represent 3-plexes with n points. Since the degree sequence does not determine a 3-plex uniquely, we must ask how many 3-plexes correspond to a given degree sequence. To do this, we again consider the complementary graph H. If the degree sequence of H is (2,2,...,2, 1,...,1, 0,...,0), a 2b n-a-2b alternative graphs arise from the grouping of the 2's into subsets that either represent disjoint cycles or interior vertices of disjoint paths. In the latter case, the subset of 2's is associated with precisely two l's. For a given integer, let Pk(a) be the number of partitions of a for which there are at most k parts that are smaller than 3. It then follows that the number of 3-plexes corresponding to the given degree sequence is b Σ Pk(a). k=0 These results do not lend themselves to the development of a simple expression for the number of 3-plexes with n points. Even worse, if we consider k-plexes for k > 3, the complementary graph is no longer very useful in the analysis. Its maximum degree will now be ≥ 3, and such graphs are far more complex than the disjoint unions of paths and cycles that the complementary graphs have been for k = 2 and k = 3. Just as the enumeration of 3-plexes presented problems that were not present when we enumerated 2-plexes, the internal structure of 3-plexes becomes far more complex than that of 2plexes. If G is a 3-plex and H = Kn - G, then if a + 2b < n (i.e., if H contains isolated points), it is easy to see that the clique graph of G is complete. When we try to go further than that statement, we see that the independent sets of H are not as easy to determine as they were when we were dealing with 2-plexes, and as a consequence the clique structure of G is much more complex. The clique structure of a k-plex for k > 3 would be still more difficult to approach. Computation of Maximal k-plexes An algorithm has been developed (Seidman, n.d.) to search for the maximal k-plexes in a given graph G. The algorithm uses depth-first search and is an elaboration of a clique-finding algorithm due to Johnston (1976). The algorithm has been programmed in PL/I using bit strings to represent the rows of the adjacency matrix of the given graph and 148 S. B. SEIDMAN AND B. L. FOSTER is included in the authors' social network analysis package (see Foster & Seidman, 1978; Seidman & Foster, 1978). IV. ILLUSTRATION A brief comparison of the network position of two prominent men in a Thai village provides an illustration of the utility of the k-plex concept. The analysis is based on concepts arising from the discussion of cliques and plexes especially the notions of each person being connected directly to every other and of robustness of structures. For our present purposes, the sociological meaning of these concepts stems from the fact that all individuals have a finite amount of time, resources, and physical energy. This places constraints on individuals' ability to carry out activities which require mobilization of many other people's support, since such mobilization cannot rest exclusively on personal contacts which are maintained and mobilized solely by the individuals' own efforts. Rather, the help of others is needed. Many considerations enter into effective mobilization of political support; basically, however, the process might be seen as consisting of two fundamental components, the first being recruitment and the second control of the ties to people who are recruited. Recruitment is essentially a matter of reachability in a graph of a suitable relation. Control, which is more complex, depends strongly on robustness of the structure induced by the individuals who have been recruited. This is so for several reasons. First, mobilization is effective only to the extent that the integrity of the set of recruits is capable of withstanding removal of some of the individuals by demographic processes (e.g., death or migration) or through recruitment by rivals. Second, "management" of the structure is facilitated by multiple channels of communication, which allow ego to keep informed of the state of the network and which allow informal pressures or social sanctions to be exerted effectively on recruits who exhibit inappropriate behavior. In terms of the plex and clique concepts, then, recruitment is associated with reachability and control with robustness of structures. The latter is lacking in diameter-n cliques; the k-plex concept, however, allows us to control both simultaneously. Briefly, the situation in the Thai village is as follows (see Foster, 1977, for details). The community is located about 25 kilometers from Bangkok on the Chao Phraya River. Although the population traditionally almost exclusively practiced wet rice cultivation, the village economy has diversified greatly in recent years as the number of full-time traders and craftsmen has increased and as many people have taken employment outside the village. Mr. Mitr (fictitious name), the headman, is a retired farmer. He is well-off by village standards, in part because of his own moderately large land holdings, but also because his two sons —inlaw live with him and have government jobs which pay well Mr. Mitr is well-liked by his fellow villagers, but he does GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 149 not exert great influence in village affairs. Mitr was born in the village and has lived virtually his entire life there. Mr. Sawasdi, on the other hand, is wealthy by almost any standard. He accumulated extremely large land holdings, both in his home village and a few kilometers away in a neighboring province, where he lived much of his life. He has a large family; all but one of his children are married, and each has already been given a large plot of land (60 rai, or about 25 acres). While living in the nearby province where much of his land is located, Mr. Sawasdi achieved a considerable amount of influence in local affairs and, in fact, was headman of a village and for several years headman of a cluster of villages, or tambon. In his present village he is more influential than the headman, as suggested by the fact that when Foster first visited the village, he was taken to Mr. Sawasdi before being taken to the headman. In view of the striking differences in these two men's background, one would expect to see a marked contrast in their place in the village social network. A comprehensive examination of the village is far beyond the scope of this paper, and we will therefore restrict our illustration to an examination of some selected features of the clique and plex structure of the friend relation (symmetrized) on the set of adult males. The data were gathered by a fixed-choice sociometric instrument as part of Foster's village census in 1971 (Foster, 1977); connected components, cliques and plexes were computed using the authors' computer programs (Foster & Seidman, 197 8; Seidman & Foster, 1978). The graph representing this relation has 162 vertices representing the adult males and other persons named by them. Thirty-nine of these points are either isolates (in this graph) or are in connected components of size two; they are of little structural interest in the present context and will be ignored. Five larger connected components exist; two contain 4 points, and two others contain 5 and 11 points respectively. The largest contains 99 points, however, including both Mitr and Sawasdi, and we will restrict our attention to that one for the rest of the discussion. The large component contains 14 non-trivial (i.e., size at least 3) cliques, one of which, has 4 points. As this information suggests, the graph of this relation is extremely complex, and visual examination yields no clear contrast. Nor is an examination of the first and second order stars of the two men of much help; Mitr is adjacent to 6 points, while Sawasdi is directly tied to 5; on the other hand, Mitr is tied to 17 people by paths of length 2, while Sawasdi is so connected to 22 others, seemingly giving a slight advantage to Sawasdi. Surprisingly, though, if the density_of the two second-order zones is computed, Mitr seems to be in the stronger position, with density of 0.14 as compared with Sawasdi's 0.11 (see Barnes, 1972, for definitions and Kapferer, 1971, for an analysis using similar concepts). If the two men's positions are compared using the principles outlined above, they contrast sharply. First, in line with the notion that an individual cannot personally form and maintain all necessary ties, we can ask which direct contacts 150 S. B. SEIDMAN AND B. L. FOSTER are most important. By definition they are individuals with whom ego has direct ties and in accordance with the principle of robustness, an effective set of first-order contacts should itself form a strong, (i.e., robust) structure—-e.g., clique(s). Mitr appears in one small clique with 3 members. Sawasdi, however, appears in 2 cliques, each with 3 members; moreover, the cliques overlap at 2 points (including Sawasdi), together forming a strong structure—i.e., a 2-plex (see Figure 2, points 87, 98, 100, and 103). Figure 2 Sawasdi's recruitment through cliques. Sawasdi is person No. 87; he is contained in two small cliques, with persons 98, 103, and 100. Note that the subgraph induced by all four points is a 2-plex. Note also that all secondary contacts depend on person 103, whose removal would disconnect all second-order contacts. We can now examine the potential of the members of the core group (i.e., the core cliques of the respective protagonists) to mobilize others. The same principles are followed as in analyzing the direct recruitment insofar as the firstorder contacts are expected to recruit effectively when they, along with the individuals they recruit, form a robust structure. Again, we search for cliques containing the recruiters, who in this case are the first-order contacts. Persons 21 and 25, Mitr's contacts, each appear in only one clique—the one including Mitr, and no effective second-order recruitment occurs. One of Sawasdi's contacts (103), however, appears in 4 cliques other than the original ones, producing the structure in Figure 2. The base of Sawasdi's power as opposed to Mitr's is now clear. The contrast becomes even more clear when we weaken our criterion of robustness slightly by allowing the first-order contacts to recruit through 2-plexes. Sawasdi's position is strengthened to include nearly twice as many individuals; just as important, his reliance on person 103 is weakened slightly by the link between persons 98 and 94, which provides a completely independent tie to the entire group of GRAPH-THEORETIC GENERALIZATION OF CLIQUE CONCEPT 151 second-order contacts. The importance of this is seen in the fact that removal of 103 from Figure 2 disconnects all secondorder ties from the first-order ties, while in Figure 3 every person remains connected (albeit weakly) to Sawasdi. Remarkably, this weakening of the robustness criterion does absolutely nothing to change Mitr's position. Figure 3 Sawasdi's recruitment through 2-plexes. Sawasdi is person 87. Person 50 appears in a clique (see Figure 2) but not in a larger 2-plex. The dotted line between persons 94 and 95 indicates that the line is not present in a 2-plex, but it appears in the subgraph induced by the total set of vertices which are in 2-plexes. It should be noted that the analysis here is motivated by theoretical considerations similar to those motivating many studies using the idea of network density. It is surprising, therefore, to find that the plex analysis produces results inconsistent with those obtained by computing the density of the second-order zones. A brief comparison of these two analyses highlights one further important property of the kplex/clique strategy. Although both in some sense focus on redundancy in the interpersonal ties, the plex strategy is able to isolate meaningful local robustness, where redundancies are socially effective. The zone density fails to discriminate between such effective, local structural properties and global properties which may well be socially meaningless (e.g., very long paths). It might seem at first glance that this difficulty could be avoided by examining density of first-order zones. Upon closer inspection, however, the same objection can be raised as for second-order zones, and in any event, assessment of levels of robustness would be completely arbitrary. V. CONCLUSIONS Fully assessing the utility of the k-plex scheme must await its further use in substantive sociological studies. Nevertheless, it is possible to compare its formal properties with those of other clique formulations. Cliques obtained 152 S. B. SEIDMAN AND B. L. FOSTER by statistical techniques are of little use in investigations of the ways in which structures can be transformed in the context of the global network. This is so since much information on the internal structure of such units and of the global graph is lost in the statistical procedures. It would be possible to regain the lost information by "superimposing" the clusters on the original network; the cliques would then be the subgraphs induced by the vertices included in the clusters. Such a procedure is still problematic, however, since the cliques have no mathematical status. The analytical utility of the k-plex notion as compared with that of true cliques stems from the fact that the extreme restrictiveness of the completeness criterion is weakened in a direct but controlled way. In addition, the graph theoretic procedure for generalizing the definition of true cliques suggests a way of studying the internal structure of empirical representatives of clique-like objects. In contrast to the kplex approach, the valued graph procedures add nothing directly to the structural aspect of true clique analysis, but by redefining relations they make the complete graph concept less restrictive. By the same token, it allows for investigation of internal structure of empirical representatives to the extent that a clique analysis of cliques can be performed. The procedure does little to illuminate more complex internal structure, however, and in itself is of no help at all in relating cliques to the global structure. The valued graph approach is in no way inconsistent with the k-plex approach and the two could be combined to achieve the advantages of both. Alba's scheme for diameter-n cliques is similar to the kplex scheme insofar as both weaken the restrictiveness of true cliques by redefining structural properties of the units rather than by redefining the relations from which they are constructed. Moreover, the internal structure of diameter-n cliques can be approached directly, but it is so little constrained that results are hard to obtain. Similarly, the relationship of diameter-n cliques to the global graph can be studied, but the extreme range of internal structural variability of even diameter-2 cliques causes intrinsic and positional properties to interact in extremely complex ways. The vulnerability of diameter-n cliques makes the global structure defined in terms of these cliques unstable. It is clear in the light of these considerations (and from the Thai village analysis) that true cliques, diameter-n cliques, k-plexes, and any of these structures constructed in valued graphs are complementary research tools. It might well be profitable to analyze the clique and/or plex structure of diameter-2 cliques or, on the other hand, the diameter-2 clique structure and plex structure of true cliques using a different relation than that from which the true cliques' were defined. Similarly, the analysis of a total network may be most profitably pursued by juxtaposing all of these and other graph theoretic structures according to a broad research strategy (see Poster & Seidman, n.d.(a), n.d.(b), for an outline of such a general network research framework). 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