Cliques, Clubs and Clans Mokken, Robert J
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www.ssoar.info Cliques, clubs and clans Mokken, Robert J. Veröffentlichungsversion / Published Version Sammelwerksbeitrag / collection article Zur Verfügung gestellt in Kooperation mit / provided in cooperation with: GESIS - Leibniz-Institut für Sozialwissenschaften Empfohlene Zitierung / Suggested Citation: Mokken, R. J. (1980). Cliques, clubs and clans. In J. M. Clubb, & E. K. Scheuch (Eds.), Historical social research : the use of historical and process-produced data (pp. 353-366). Stuttgart: Klett-Cotta. https://nbn-resolving.org/ urn:nbn:de:0168-ssoar-326164 Nutzungsbedingungen: Terms of use: Dieser Text wird unter einer Deposit-Lizenz (Keine This document is made available under Deposit Licence (No Weiterverbreitung - keine Bearbeitung) zur Verfügung gestellt. Redistribution - no modifications). We grant a non-exclusive, non- Gewährt wird ein nicht exklusives, nicht übertragbares, transferable, individual and limited right to using this document. persönliches und beschränktes Recht auf Nutzung dieses This document is solely intended for your personal, non- Dokuments. Dieses Dokument ist ausschließlich für commercial use. All of the copies of this documents must retain den persönlichen, nicht-kommerziellen Gebrauch bestimmt. all copyright information and other information regarding legal Auf sämtlichen Kopien dieses Dokuments müssen alle protection. You are not allowed to alter this document in any Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen way, to copy it for public or commercial purposes, to exhibit the Schutz beibehalten werden. Sie dürfen dieses Dokument document in public, to perform, distribute or otherwise use the nicht in irgendeiner Weise abändern, noch dürfen Sie document in public. dieses Dokument für öffentliche oder kommerzielle Zwecke By using this particular document, you accept the above-stated vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder conditions of use. anderweitig nutzen. Mit der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an. Robert J. Mokken Cliques, Clubs and Clans* 1. Introduction In the analysis of social networks adequate concepts are necessary to indicate various types and configurations of social groups such as peer groups, coteries, acquaintance groups, etc. The problem has theoretically been argued convincingly a. o. by Kadushin, who introduced the general concept of „social circle**1. In the actual empirical study of social networks there is therefore a need for adequate operational and analyticaUy useful concepts to represent such more or less closely knit groups. Many of these can be developed with the help of the theory of graphs and net¬ works. A weU-known concept, more or less corresponding to that of the peer group is the clique: a group aU members of which are in contact with each other or are friends, know each other, etc. However, similar concepts wül be necessary to de¬ note less closely knit, yet significantly homogeneous social groups, such as „acquain¬ tance groups'*, where every pair of members, if they are not in mutual contact, have mutual acquaintances, or common third contacts, etc. In this latter type of social group an important aspect is given by the question whether the homogeneity of a social group is due to its position in a larger social network in which it is embedded, or whether it is a property of the group itself as a more or less autarchic unit, inde¬ pendent of the surrounding social network. In the first case, for instance, a group may be as closely knit as an „acquaintance network**, just because there are „mu¬ tual acquaintances'* outside the group in the surrounding network. Changes in the environment, i. e. the outside network, may change the character of the group as an acquaintance group. In the latter case, however, a group is an acquaintance group, because mutual acquaintances linking members are themselves members of that group. Therefore changes in the outside social network wül not effect the nature and structure of the group itself as an acquaintance group. Such concepts can be * Part of this research was performed during the authors stay as Visiting Professor at the Uni¬ versity of Michigan. The author owes much to the stimulating discussions with Professor Frank Harary. 1 Kadushin, Charles, Power, Influences and Social Circles: A new methodology for studying opinion makers, in: American Sociological Review, 33 (1968), pp. 685—699. 353 worked out in terms of graph theoretic cluster concepts. For instance, the famüiar concept of n-clique deals with the tightness of a group as a global property, due to the interrelationships or interactions of all members of a larger social network. The concepts of clubs and clans, to be introduced here denote a local property of struc¬ tural autarchy in the sense that the interrelationships within the particular social group are sufficient for its homogeneity, and independent of those interrelation¬ ships involving members or parts of the surrounding larger social network. In this paper we shall introduce three different cluster concepts of graphs - cliques, clubs and clans — and investigate their interrelationships. The graphs treat¬ ed here wiU be simple graphs: finite, nonempty, and having no loops or multiple lines. We shall mainly follow the notation and concepts given by Harary, to which we may refer the reader for further reference2. We shall be satisfied here with a cur- sory introduction of the concepts and notation used here. A graph G is a set of points together with a set of lines. To simplify notation here, we shaU use the same symbol G to denote the set of points of G. Any line of G connects some pair of points u, v € G, which then are said to be adjacent to each other in G. We shaU also consider subgraphs of G, indicated by their pointset. If H C G is a subset of G, the subgraph H of G consists of all points of H together with aü lines of G, which connect points u, v e H in G. A path, connecting two points u, v a that same consists of . of subgraph H, in subgraph H, points u, w^, W2, w^_ j veH, such that u is adjacent to w^, w^is adjacent to Wj+j, consecutively and w-_i is adjacent to v. The length / of a path is given by the number of its lines. A cycle C- of length / is apath of length /, where u = v. A subgraph H is connected in G, if each pair of points u, v e H is connected by a path in H. A complete graph Kp is a graph of p points, where each pair of points is adjacent to each other. We shall also consider maximal subgraphs with respect to a given property. They are subgraphs of G satisfying that property, such that no larger subgraphs with that property exist in G, which contain them. A well known example is given by the cUques of a graph G: maximal complete subgraphs Kp of G. The distance of a pair of points u, v in a certain subgraph H, denoted by dH (u, v) is given by the length of a shortest path connecting u and v in H. If u, v are not connected in H, this distance is infinity. We shall frequently make use of the weU-known relation that, if H is a subgraph of G, than for every pair of points u, v e H dG (u, v) < dH (u, v) (1) The distance between any two points in a subgraph of G cannot be smaller than their distance in G itself. The diameter of a subgraph H is given by the largest distance between a pair of points in that subgraph. 2 Harary, F., Graph Theory, Reading/Mass. 1969. 354 c we If we extend the pointset H C G with a point w € G—H or a subset S G—H, in the to H u or may consider the distances larger subgraphs corresponding { w} for distances as or The of a u Hu S, denoting simplicity dpj w dj-j g.* degree point in G is the number of points (neighbors) adjacent to u in G. The set of those points is calied the 1-neighborhood of u in G, to be denoted by V (u). We may restrict the set of neighbors to those in a subgraph H of G only, to be denoted as V (u). Similar extensions may be made to n-neighborhoods of u: points at distance n of u. 2. Cluster Concepts In Standard graph theory a familiär Cluster concept is given by the cliques of a graph G. As mentioned above, they are given by the set of maximal complete subgraphs of G. Another Cluster type definition of subgraphs of graphs are the ,,n-cliques", in¬ troduced by Luce as given by the following definitions.** Definition 1. An n-clique L of a graph G is a maximal subgraph of G such that for all pairs of points u, v of L the distance in G dG(u,v)<n. (2) The reader may note that, due to the maximality of L, for every point w e G—L, there is a point v e L for which dG (w, v) > n (3) From this definition, it can be seen that the n-clique is a global concept, based on the total structure of the network, as based on the graph and reflected in its distance matrix. The distances between points in a certain subset of points, can be based on shortest paths, involving other points from the network not belonging to that group. It is weU-known, therefore, that in the subgraph, formed by the points of an n- clique L, the distances between points can be larger than n. This follows from the *The conventional notation of set theory is used. In particular S^G wül denote set inclusion, = S C C proper inclusion and S G identity of sets.