Structural Cohesion and Embeddedness: a Hierarchical Concept of Social Groups Author(S): James Moody and Douglas R

Structural Cohesion and Embeddedness: A Hierarchical Concept of Social Groups Author(s): James Moody and Douglas R. White Source: American Sociological Review, Vol. 68, No. 1 (Feb., 2003), pp. 103-127 Published by: American Sociological Association Stable URL: http://www.jstor.org/stable/3088904 . Accessed: 02/02/2015 19:14

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This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS: A HIERARCHICAL CONCEPT OF SOCIAL GROUPS

JAMES MOODY DOUGLAS R. WHITE The Ohio State University University of California-Irvine

Although questions about social cohesion lie at the core of our discipline, definitions are often vague and difficult to operationalize. Here, research on social cohesion and social embeddedness is linked by developing a concept of structural cohesion based on network node connectivity. Structural cohesion is defined as the minimum numberof actors who, if removedfrom a group, would disconnect the group.A structural dimension of embeddedness can then be defined through the hierarchical nesting of these cohesive structures. The empirical applicability of nestedness is demonstrated in two dramatically different substantive settings, and additional theoretical implications with reference to a wide array of substantive fields are dis- cussed.

"[Slocial solidarity is a wholly moral phenom- of "community cohesion" for preventing enon which by itself is not amenable to exact ob- crime (Sampson and Groves 1989). Political servation and especially not to measurement.P" sociologists focus on how a cohesive civil -Durkheim ([1893] 1984:24) society promotes democracy (Paxton 1999; Putnam 2000). Historical sociologists point "The social structure [of the dyad] rests immedi- to the importance of solidarity for revolu- ately on the one and on the other of the two, and tionary action (Bearman 1993; Gould 1991), the secession of either would destroy the and that the success of heterodox social whole.... As soon, however, as there is a sociation of three, a group continues to exist even movements depends on a cohesive critical in case one of the members drops out." mass of true believers (Oliver, Marwell, and Teixeira 1985). Social epidemiologists argue -Simmel ([1908] 1950:123) that a cohesive "core" is responsible for the persistence of sexually transmitted diseases Q UESTIONS SURROUNDING social (Rothenberg, Potterat, and Woodhouse solidarity are foundational for soci- 1996). Worker solidarity is a key concept in ologists and have engaged researchers con- the sociology of work (Hodson 2001). So- tinuously since Durkheim. Researchers cial psychologists have repeatedly returned across a wide spectrum of substantive fields to issues surrounding cohesion and solidar- employ "cohesion" or "solidarity" as a key ity, attempting to understandboth its nature element of their work. Social disorganization (Bollen and Hoyle 1990; Gross and Martin theorists, for example, tout the importance 1952; Roark and Shara 1989) and consequences (Carron 1982; Hansell 1984). Direct all correspondence to James Moody, 372 Bricker Hall, Department of Sociology, The Ferrand, Frank Harary, David Jacobs, David R. Ohio State University, Columbus, OH 43210 Karger, Lisa Keister, Mark Mizruchi, Scott (Moody.77 @sociology.osu.edu). This research Provan, Sandeep Sen, Mechthild Stoer, and three was funded by NSF grants SBR-9310033 and anonymous reviewers for help and comments. BCS-9978282 to Douglas White. The first grant We also thank the following members of the was matched by an Alexander von Humboldt Santa Fe Institute Working Group on Co-Evolu- Transatlantic Cooperation Award to Thomas tion of Markets and the State: Sanjay Jain, John Schweizer, University of Cologne. We thank Pe- Padgett, Walter Powell, David Stark, and Sander ter Bearman, Steve Borgatti, Bob Farris, Alexis van der Leeuw.

AMERICAN SOCIOLOGICAL REVIEW, 2003, VOL. 68 (FEBRUARY: 103-127) 103

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Unfortunately, as with "structure" smaller director-interlocknetwork, we show (Sewell 1992), the rhetorical power of "co- that joint network embeddedness leads dy- hesion" is both a blessing and a curse. Soci- ads to make similar political contributions, ologists are all too familiar with the prob- linking network position to coordinated po- lem: We study "cohesion" in almost all our litical action. In both cases, we find indepen- substantive domains, and in its ambiguity, it dent effects for our conception of cohesion seems to serve as a useful theoretical place- net of commonly used alternative measures, holder. Ubiquity, however, does not equal substantiatingits unique contribution. theoretical consistency. Instead, the exact meaning of cohesion is often left vague, or when specified, done in a particularistic BACKGROUND AND THEORY manner that makes it difficult to connect in- SCOPE sights from one subfield to another. We identify one generalizable structuraldimen- Analytically, solidarity can be partitioned sion of social solidarity. Although the con- into an ideational component, referring to cept we develop is related in certain ways members' identification with a collectivity, to some, perhaps many, of the meanings of and a relational component (Dorian and "solidarity" or "cohesion" used in the litera- Fararo 1998), referring to the observed con- ture, it is by no means intended to incorpo- nections among members of the collectivity. rate them all. Instead, we focus on only one This theoretical distinction, for example, al- dimension. By carefully identifying one as- lowed Durkheim to link changes in the com- pect of social solidarity, we hope to help mon consciousness to the transition from clarify one of the multiple meanings con- mechanical to organic societies, although he tained in this ubiquitous idea. offered no clear measures for these concepts. The social network-based concept we de- Research on commitment (Kanter 1968) or velop is theoretically grounded in insights perceived cohesion (Bollen and Hoyle 1990) from Simmel ([1908] 1950) and Durkheim focuses directly on the ideational component ([1893] 1984) and methodologically of social solidarity. Although often based on grounded in classical graph theory (Harary an underlying relational theory, much of the 1969; Harary, Norman, and Cartwright national-and community-level work on 1965). D. White and Harary (2001) demon- social cohesion uses ideational indicators of strate the formal logic by which graph-theo- "community cohesion" (Paxton 1999; retic measures lend themselves to the study Sampson and Groves 1989). Distinguishing of the structural dimension of social cohe- between the relational and ideational com- sion. Here, we extend a definition of struc- ponents analytically does not imply a causal tural cohesion in its most general form, ap- precedence of one dimension over the other. plicable to large-scale analyses in a variety Empirically, these two dimensions (and per- of settings, and provide an algorithm for its haps others) might mutually reinforce each use in empirical analyses. The implementa- other. Whatever their ultimate causal relation of our algorithm for measuring embed- tion, separating these two dimensions is a ded levels provides an operational specifica- prerequisite to identifying the relation be- tion of one dimension of social embedded- tween them. Here we leave the wider ques- ness (Granovetter 1985, 1992), which allows tion of "social solidarity" in the background us to specify and explore empirically the and focus instead on structural cohesion: a unique contribution of this dimension. Here single dimension of the relational compo- we focus on two empirical settings: friend- nent of social solidarity. ships among high-school students (Bearman, Some of the ambiguity surroundingappli- Jones, and Udry 1996) and the political ac- cations of "cohesion" and research on cohe- tivity of big businesses (Mizruchi 1992). For sive groups involves differences in scale. adolescent friendships, we show that net- Although the theoretical importance of so- work position predicts school attachment, cial cohesion is often cast at national levels using structural cohesion to link the rela- (Durkheim [1893] 1984; Putnam 2000), tional to the ideational components of soli- most treatmentsof the relational dimensions darity in a dozen large networks. For the of cohesion have focused on small groups.

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Structuralcohesion, however, is no less im- ued presence is requiredto retain the group's portant at larger scales, although the rela- connectedness (for graph theoretical aspects, tional connectivities that might define cohe- see D. White and Harary 2001). For clarity sion cannot be equally dense. An advantage and theoretical consistency, we refer to this of our concept of structuralcohesion is that relational aspect of social solidarity as struc- it applies to groups of any size.' In so doing, tural cohesion. Structuralcohesion simulta- we add a new dimension to recent literature neously defines a group property character- on large-scale social networks (Barabasi and izing the collectivity, a positional property Albert 1999; Newman 2001; Watts 1999) that situates subgroups relative to each other and bridge insights about small-group struc- in a population, and individual membership ture to those at much larger scales. properties. Although we do not claim to cap- Identifying cohesive structuresis only one ture the full range of either "solidarity" or part of analyzing structuralcohesion, and a "embeddedness," structural cohesion pro- more informative approach simultaneously vides an exact analytic operationalization of tells us how such groups relate to one an- a dimension of each. other. Our concept of structural cohesion necessarily entails a positional analysis of DEFINING STRUCTURAL COHESION the resulting groups with respect to their nesting within the population at large. Theo- Research on social cohesion has been retically, the resulting concept of nestedness plagued with contradictory,vague and diffi- captures one dimension of Granovetter's cult-to-operationalize definitions. Mizruchi (1985, 1992) concept of social embedded- (1992, chap. 3) provides a useful discussion ness. Like "solidarity,""embeddedness" is a of the conflation of "sharednormative senti- multidimensional construct relating gener- ments" and "objective characteristics of the ally to the importance of social networks for social structure" in definitions of cohesion action. Embeddedness indicates that actors (also see Doreian and Fararo 1998; Mudrack who are integrated in dense clusters or mul- 1989). Many of these definitions share only tiplex relations of social networks face dif- an intuitive core that rests on how well a ferent sets of resources and constraints than group is "held together."What does it mean, those who are not embedded in such net- for example, that cohesion is defined as a works. By specifying an exact structuralin- "field of forces that act on members to re- dicator for one dimension of social embed- main in the group" (Festinger, Schachter, dedness, we move beyond orienting state- and Back 1950) or "the resistance of a group ments and augment our ability to develop to disruptive forces" (Gross and Martin cumulative scientific insights. 1952)? Dictionary definitions of cohesion Here, we identify an important feature of rest on similar ambiguities, such as "[t]he the relational dimension of social solidarity action or condition of cohering; cleaving or that is applicable to groups of any size. Fol- sticking together" (Oxford English Dictio- lowing Simmel, that feature is the extent to nary 2000). Although we might all agree that which a group depends on particular indi- cohesive groups should display "connected- viduals to retain its characteras a group. The ness" (O'Reilly and Roberts 1977), what as- relevant quantitative measure is the mini- pects of connectedness should be taken into mum number of individuals whose contin- account? For concepts of cohesion to be analytically 1 Because of the long history of small face-to- useful, we must differentiate the relational face research on groups, we prefer to avoid the togetherness of a group from the sense of to- use of the term "groups" altogether, in favor of getherness that people express. Using only broader terms such as "collectivity" or "substruc- subjective factors, such as a "sense of we- ture" that carry much less theoretical baggage. ness" (Owen 1985) or "attraction-to-group" Such a substitution, however, results in decidedly (Libo 1953), fails to capture the collective awkward writing. We thus maintain the use of of a cohesive ''group," but remind readers that our concept is nature group (Mudrack 1989). not limited to the small face-to-face primary Conversely, many treatments that focus ex- group structures commonly referred to by the clusively on groups, such as the group's abil- term. ity to "attract and retain members," com-

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 106 AMERICAN SOCIOLOGICAL REVIEW mingle the relational and ideational compo- tions.3 This intuition is captured well by nents of social solidarity. Conflating rela- Markovsky and Lawler (1994) when they tional and ideational features of social soli- identify "reachability"as an essential feature darity in a single measure limits our ability of relational cohesion. Additionally, as new to ask questions about how the relational relations form within this minimally cohe- component of solidarity affects, or is af- sive group, we can trace multiple paths fected by, ideational factors. through the group. Intuitively, the ability of The ability to directly operationalize the group to "hold together" increases with structural cohesion through social relations the number of independent ways that group is one of the primary strengths of a relational members are linked. conception of social cohesion. The "forces" That cohesion seems to increase as we add and "bonds"that hold the group together are relations among pairs has prompted many the observed relations among members, and researchers to focus on the volume (or den- cohesion is an emergent property of the re- sity) of relations within and between groups lational pattern.2 (Alba 1973; Fershtman 1997; Frank 1995; Based on this prior literature, a prelimi- Richards 1995). There are two problems nary intuitive definition of structural cohe- with using relational volume to capture sion might read: structural cohesion in a collective. First, consider again our group with one path con- Definition 1: A collectivity is structurally necting all members. We can imagine mov- cohesive to the extent that the social re- ing a single relation from one pair to another. lations of its members hold it together. In so doing, the ability to trace a path be- While we will sharpen the terms of defini- tween actors may be lost, but the number of tion 1 below, there are five important fea- relations remains the same. Because volume tures of this preliminary definition. (1) It fo- does not change but reachability does, vol- cuses on what appears to be constant in pre- ume alone cannot account for structuralco- vious definitions of cohesion: A propertyde- hesion. scribing how a collection of actors is united. Second, the initial (and weakest) moment (2) It is expressed as a group-level property. of structural cohesion occurs when we can Individuals may be embedded more or less trace only one path from each actor in the strongly within a cohesive group, but the network to every other actor in the network. group has a unique level of cohesion. (3) Now imagine that our ability to trace a chain This concept is continuous. Some groups from any one person to another always will be weakly cohesive (not held together passes through a single person. This might well), while others will be strongly cohe- occur, for example, if all relations revolved sive. (4) Structuralcohesion rests on observ- around a charismatic leader: Each person able social relations among actors. And (5), might have ties to the leader, and be con- the definition makes no reference to group nected only through the leader to every other size. member of the group. While connected, such What, then, are the relational features that groups are notoriously fragile. As Weber hold collectivities together? Clearly, a col- ([1922] 1978:1114) points out, the loss of a lection of individuals with no relations charismatic leader will destroy a group among themselves is not cohesive. If we whose structureis based on an all-to-one re- imagine relations forming among a collec- lational pattern. Thus, increasing relational tion of isolated individuals we might observe volume but focusing it through a single in- a moment when each person in the group is dividual does not necessarily increase the connected to at least one other person in ability of the group to hold together, and in- such a way that we could trace a single path from each to the other. Thus, a weak form of structuralcohesion starts to emerge as these 3See Hage and Harary (1996) for a discussion islands become connected through new rela- of this process among islands in Oceania. We recognize that social groups can form from the dissolution of past groups; the above discussion 2 This concept assumes that the dyadic relation is useful only in understanding the character of is a positive connection. structuralcohesion.

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS 107 stead makes the network vulnerable to tar- social characteristics. Groups of any size geted attack.4 that depend on connections through a single Markovsky and Lawler (1994; Markovsky actor are at one end of definition 2 (weakly 1998) make a related point when they argue cohesive), while those that rest on connec- that a uniform distribution of ties is needed tions through two actors are stronger, and to prevent a network from splitting into mul- those depending on connections through tiple subgroups: many actors are stronger yet. The strongest cohesive groups are those in which every [T]he organization of [cohesive] group ties person is directly connected to every other should be distributed throughout the group person (cliques), though this level of cohe- in a relatively uniform manner. This implies the absence of any substructures that might sion is rarely observed except in small pri- be vulnerable, such as via a small number of mary groups.5 Intuitively similar ideas moti- "cut-points" to calving away from the rest vated earlier measures of social cohesion in of the structure. (Markovsky 1998:345) networks, such as Seidman and Foster's (1978) treatment of k-plexes (Seidman Such vulnerable substructures form when 1983). They argue that a key feature of network relations are focused through a cliques that needs to be preserved in any small number of actors. If pairs of actors are measure of group cohesion "is the robust- linked to each other through multiple others, ness of the structure. This property ... is the structureas a whole is less vulnerable to best characterized with reference to the de- this type of split. gree to which the structure is vulnerable to Given the above, we amend our prelimi- the removal of any given individual" (p. nary definition of structural cohesion to 142). The k-plex characterization, however, make explicit the importance of multiple in- cannot ensure that this lack of vulnerability dependent paths linking actors together: is achieved. To specify our concept of structuralcohe- Definition 2: A group is structurallycohesive sion, we need a language capable of accu- to the extent that multiple independent rately expressing relational patterns in a relational paths among all pairs of mem- group. The language of graph theory pro- bers hold it together. vides this clarity. Because graph-theoretical While still preliminary, this new definition terms are unfamiliar to many, however, we provides a metric for structuralcohesion that illustrate the definitions below with refer- reflects Simmel's ([1908] 1950:135) discus- ence to Figure 1 following the definition. A sion of the supra-individual status of triads network is composed of actors, represented over dyads. In a dyad, the existence of the as nodes in the graph, and the relations group rests entirely in the actions of each among them, represented as edges. Struc- member, as either member acting unilater- tural cohesion depends on how pairs of ac- ally could destroy the dyad by leaving. Once tors can be linked through chains of rela- we have an association of three, however, a tions, or paths. A path in the network is de- connected group remains even if one of the fined as an alternating sequence of distinct members leaves. In triads, the social unit is nodes and edges, beginning and ending with not dependent on a single individual, and nodes, in which each edge is incident with thus the social unit takes on new uniquely its preceding and following nodes (Figure la, {1-2->5->6}). We say that actor i can 4 Recent research on large networks such as the reach actor j if there is a path in the graph World-Wide Web finds that an extremely small starting with i and ending with j (Figure la, number of nodes are connected to an extremely 1 can reach 7 {1->3->6->7}, but 1 cannot large number of partners.These networks depend reach 11). Two paths from i to j are node- on high-volume actors to remain connected, and independent if they have only nodes i and j targeted interventions (virus attacks in computer networks, education and treatment effects in sexually transmitted disease networks) will dis- I It is important that our measure of structural connect the network and disrupt flow (Barabasi cohesion reaches a maximum with fully con- and Albert 1999; Pastor-Satorrasand Vespignani nected cliques, linking us to previous concepts of 2001). network cohesion.

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(a) (b) (c) (d)

k =1 k =1 k =2 k =3 Figure 1. Examples of Connectivity Levels in common (Figure 1a,{ 1-3--6} and must have at least two nonadjacentnodes 1-2--5--6} are node independent but connectedby paths, all of which must pass {1o-2-+5-+6} and {1o-2-+7-+6} are not througha cutset of k othernodes (in Figure node independent).If thereis a pathlinking Ic thereare two suchpaths connecting 3 and all pairs of actors in the network,then the 13: {3-+6--11--12--13} and {3-+9--13}). graphis connected(Figures lb throughId). Whatis not so obvious, constitutingone of A componentof a networkconsists of all the deepesttheorems about graphs, is that a nodesthat can be connectedto each otherby k-connectedgraph (i.e., havinga cutsetwith at least one path.Components are the mini- exactlyk members)also has at least k node- mum setting for a cohesive structure.A independentpaths connectingevery pair of cliqueis a maximallyconnected subgraph in nodes, and vice versa (see Harary1969 for whichevery member is directlyconnected to Menger'sproof).7 every other member (Figure la, { 12, 13, 141). A outsetof a graphis a collection of not the variantusage, thatlends itself to discus- specific nodesthat, if removed,would break sion of cohesionin termsof k-components. the componentinto two or morepieces (Fig- 7 D. White and Harary(2001) formalize the ure lb, node 7; Figure ic, nodes 6 and 13). definitionof structuralcohesion, review the cri- A graph is k-connected (i.e., has node con- tiques of alternatemeasures of cohesive sub- nectivity k) and is called a k-component if it groups, and then discuss the relation between has no cutsetof fewer thank nodes (Harary connectivityand density. They also examine a second but weakerdimension upon which such 1969:45-46). In graph-theoreticterminol- groupscould be arrangedthat relates to edge con- ogy, a 2- or biconnectedcomponent is called nectivity(also see Borgatti,Everett, and Shirey a bicomponent (Figure ic) and a 3-con- 1990;Wasserman and Faust 1994), measuredby nected componenta tricomponent(Figure the minimumnumber of edges that must be re- ld).6 Any k-componentis eithera clique or movedin a connectedgroup that will resultin its disconnection.It can be shown that a graphof 6 In Harary's terminology, for example, an iso- any level of edge connectivitymay still be sepa- lated pair of connected individuals is not 2-con- rableby removalof a single actor,which means nected and do not constitute a bicomponent. The that the unilateralpower of actors can be high algorithmic and computer science literature con- even when there are many relationsconnecting stitute a variant usage in which a k-component is people. We differ from D. White and Harary any graph that cannot be disconnected by re- (2001) by generalizingthe theoreticallinks to moval of fewer than k nodes, hence a bicom- social solidarity, developing the link between ponent, for example, includes an isolated pair of nestednessand embeddedness,providing an al- connected individuals. It is Harary's usage and gorithmto facilitateempirical research using co-

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Based on the intuitive notions captured in differing levels of structuralcohesion. Note Definition 2 and the formal graph properties that in each of these three groups the num- presented above, we can now provide a final ber of relations is held constant, but the definition of structuralcohesion: edges are arrangedsuch that structuralcohesion increases from left to right. Definition 3a: A group's structuralcohesion is equal to the minimum number of actors who, if removed from the group, COHESIVE BLOCKING would disconnect the group. An algorithm for identifying structurallyco- A group is cohesive to the extent that it is hesive groups is described in Appendix A. robust to disruption, which is captured by Identification involves a recursive process: node connectivity. For each connectivity One first identifies the k-connectivity of an value (k) observed in a given network, there input graph, then removes the k-cutset(s) that is a unique set of subgroups with this level hold(s) the network together. One then re- of structuralcohesion. Because of the formal peats this procedure on the resulting sub- equality between the size of the cutset and graphs, until no further cutting can be done. the number of node-independent paths, the As such, any (k+l)-connected set embedded "disconnect" version of Definition 3a can be within the network will be identified. More- restated without any loss of meaning in "held over, each iteration of the procedure takes us together" terms as: deeper into the network, as weakly connected nodes are removed first, leaving stronger and Definition 3b: A group's structuralcohesion stronger connected sets, uncovering the is equal to the minimum number of in- nested structureof cohesion in a network. dependent paths linking each pair of ac- This search procedure can result in two tors in the group. types of subgroups. On one hand, we may This pair of equivalent definitions of identify groups that "calve away" from the structuralcohesion retains all five aspects of rest of the population, such as those dis- our original intuitive definition of structural cussed by Markovsky and Lawler (1994). In cohesion. A collection of actors is united such cases, cohesive groups rest "side-by- through relational paths that bind nodes to- side" in the social structure, one distinct gether. Node connectivity is a group-level from the other. This is the kind of descrip- property (a network or k-component as a tion commonly used for primary social whole is k-connected), but individuals can be groups (Cooley 1912), which we expect to more or less strongly embedded within the exhibit high levels of structural cohesion. group (as the network may admit to nested Alternatively, structurally cohesive groups (k + 1)-connected subgroups). This concep- could be related like Russian dolls - with in- tion scales, ranging from 0 (not connected) creasingly cohesive groups nested inside to n - 1 (a complete clique), and applies to each other. The most common such example networks of any size. would be a group with a highly cohesive Structural cohesion is weaker to the de- core, surrounded by a somewhat less cohe- gree that connectivity depends on a small sive periphery, as has been described in number of actors. Such graphs are vulner- widely ranging contexts (Borgatti 1999). A able to the activity of fewer and fewer mem- common structuralpattern for large systems bers. As node connectivity increases, vulner- might be that of hierarchical nesting at low ability to unilateral action decreases. Based connectivity levels and nonoverlapping on Simmel's discussion of the dyad, we ar- groups at high connectivity. gue that a connectivity of 2 (a bicomponent) To gain an intuitive sense for the cohesive is the minimum distinction between weak group detection procedure, consider the ex- and stronger structurally cohesive groups, ample given in Figure 2. This network has a which are ranked by their k-connectedness. single component inclusive of all nodes. Figure 1 presents examples of networks with Embedded within this network are two biconnected components: nodes {1-7, 17- hesive blocking, and expanding the empirical ap- 23) and {7-16}, with node 7 involved in plication of these measures to new settings. both. Within the first bicomponent, however,

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Figure 2. Nested Connectivity Sets members {1-7} form a 5-componentand thanone maximalk-cohesive group, that in- members {17-23} form a 3-component. dividualis partof a uniquesubset of k - I or Similarly,nodes {7, 8, 11, and 14} forman- fewer individualswhose removal will dis- other 3-component(a four-personclique) connect the two groups. Membersof such within the second bicomponent,while the bridging sets are positionally equivalent remainderof the groupcontains no sets more with respectto the largercohesive sets that strongly connected than the bicomponent. they bridge.9 As such, a positional and rela- Thus, the group structureof this network tional structurecomes out of the analysisof contains a 3-level hierarchy,which is pre- cohesive groups. These groups are much sentedin Figure3. larger,fewer, and easier to distinguishthan Because connectivity sets can overlap, aretraditional sociological cliques. This pro- group members can belong to multiple cedureprovides some of the same theoreti- groups.Although observed overlaps at high cal purchase that blockmodels were de- levels of connectivitymay be rare,any ob- are sig- servedoverlaps likely substantively tionship patterns. Arguments that relative density nificant.8If an individualbelongs to more groups (cf. Frank 1995) solve this problem by assigning each actor to their preferred group 8 Some researchersconsider overlapping sub- (based on number of nominations) fail to account groupstoo empiricallyvexing to provideuseful for people who have ties across many subgroups, analysis.It is importantto point out that (1) k- such that the total number of ties to people in componentsare strictly limited in the size of such other groups is higher than the number of ties to overlaps,making the substantivenumber of such people in the group they have been assigned to. intermediatepositions small-especially com- 9 In Figure 2, for example, only the removal of paredwith cliques; (2) that each such position, node 7 will disconnect the 1-component, removal because of its known relation to the potential of {5, 7},{21, 19}, {5, 19},{21, 7} will discon- flow paths and cycle structureof the network, nect the bicomponent along similar lines, while can be theoreticallyarticulated in ways that are removal of only {8, 10} will disconnect 9 from impossible for clique overlaps; and (3) even the rest, only {14, 16} will disconnect 15 from whenoverlaps are empirically difficult to handle, the rest, and only { 10, 16} will disconnect 13 they may well be an accuratedescription of rela- from the rest, and so forth (see Appendix A).

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{7,8,9,10, 11 {1,2,3,4,5,6,7, 12,13,14,15,16) 17,18,19, 20, 21, 22, 23)

{7,8,11,14} {1,2,3,4, {17,18,19,20, 5,6,7) 21,22,23)

Figure 3. Cohesive Blocking for the Network in Figure 2

signed to provide (Burt 1990; Lorrain and vetter 1985, 1992). The generalconcept of White 1971; H. White, Boorman, and embeddednesshas had a significant influ- Breiger 1976), but focuses on subgraphsthat ence in currentsociological research and may overlap rather than on partitions of theory. Although used most often in eco- nodes.'0 Because this method provides the nomic sociology (Baum and Oliver 1992; ability to both identify cohesive groups and Portesand Sensenbrenner1993; Uzzi 1996, identify the position of each group in the 1999) or stratification(Brinton 1988), em- overall structure, we call the method cohe- beddednesshas been used to describesocial sive blocking. It is importantto note the flex- support (Pescosolido 1992), processes in ibility of this approach. The concept of co- healthand healthpolicy (Healy 1999; Ruef hesion presented here provides a way of or- 1999), family demography(Astone et al. dering groups within hierarchically nested 1999), andthe analysisof criminalnetworks trees, with traditional segmented groups oc- (Baker and Faulkner 1993; McCarthy, cupying separate branches of the cohesion Hagen, and Cohen. 1998). Most treatments structure (recall Figure 3), but allowing of embeddednessrefer to the constraining overlap between groups in different effectsof social relations,contrasting "arms- branches (e.g., node 7 in Figure 3). The abil- length relations"(Uzzi 1996, 1999) or "at- ity to accommodate both nested and seg- omized individuals" (Granovetter 1985, mented structureswithin a common frame is 1992) to action that is embeddedin social a strength of our model. relations." Embeddednessis often used to claim a broadorientation to theoriesof social action,delineating a spacefor actionbe- RELATION OF COHESION TO tween "undersocialized"perspectives that SOCIAL EMBEDDEDNESS treatactors as completelyindependent util- A nested concept of cohesion provides a di- ity maximizers and "oversocialized"per- rect link between structuralcohesion and an spectivesthat treat actors as culturaldupes. element of social embeddedness (Grano-

10 Overlaps are crucial to cohesive structures. I "Embeddedness" suffers from some of the Ordinarily we think of social groups as designa- same ambiguity evident in discussions of solidar- tions for sets of individuals. Structural cohesion ity and cohesion, with the term being used to de- identifies groups in terms of sets of relationships, note many different aspects of the importance of as represented by edges in the graph. For ex- relationships for social action. While some re- ample, bicomponents may result in a partition of searchers have keyed their definition of embed- edges (not individuals) allowing people to be in dedness directly to a given type of action out- multiple cohesive groups. It is for this reason that come, we prefer a concept of embeddedness that cohesive blocking cannot generally be subsumed lets us test whether a particular pattern of rela- as a form of blockmodeling: Cohesive blocks tions shapes action, instead of defining embed- may overlap and do not form partitions. dedness in terms of the resulting constraint.

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To move from orienting statement to em- pect of structuralembeddedness-the depth pirical investigation, we must identify clear of involvement in a cohesive structure-is dimensions of embeddedness that would ad- captured by this nesting. We define an mit to empirical operationalization. Grano- actor's nestedness in a social network as the vetter (1992) points to a key division be- deepest outset level within which the actor tween "local" and "structural" embedded- resides.12 ness:

"Embeddedness" refers to the fact that eco- ALTERNATIVE APPROACHES TO nomic action and outcomes, like all social STRUCTURAL COHESION action and outcomes, are affected by actors' dyadic (pairwise) relations and by the struc- Node connectivity differs markedly from ture of the overall network of relations. As a other approaches to identifying cohesive shorthand, I will refer to these as the rela- groups in social networks.13While previous tional and the structural aspects of embed- work on structural cohesion was motivated dedness. The structural aspect is especially by questions of graphvulnerability (Seidman crucial to keep in mind because it is easy to 1983; Seidman and Foster 1978), the result- slip into "dyadic a of re- atomization," type ing empirical measures could not ensure a ductionism. (P. 33, italics in original) nonvulnerable graph. Group identification Granovetter (1992) further specifies his un- methods based on numberof interactionpart- derstanding of structural embeddedness as ners (k-plexes, k-cores), minimum within- the degree to which actors are involved in group distance (N-cliques), or relative in- cohesive groups: group density, may be structurallycohesive, [T]o the extent that a dyad's mutual contacts but are not necessarily so. In every case, the are connected to one another, there is more method used to identify groups cannot dis- efficient information spread about what tinguish multiconnected groups from those members of the pair are doing, and thus bet- vulnerable to the removal of a single actor. ter ability to shape behavior. Such cohesive As such, any empirical application of these groups are better not only at spreading in- methods to a theoretical problem of struc- formation, but also at generating normative, tural cohesion risks ambiguous findings. By symbolic, and cultural structures that affect distinguishing structuralcohesion from fac- our behavior." (P. 35) tors such as density or distance, we can iso- Granovetter's concept invokes transitivity late the relative importance of connectivity (Davis 1963; Holland and Leinhardt 1971; in social relations from these other factors. Watts 1999), focusing on the pattern of rela- Distance between members, the number of tions among a focal actor's contacts. One common ties, and so forth might affect out- need not limit structuralembeddedness to an comes of interest, but our ability to extend actor's direct neighborhood, however, but social theory in formal network terms de- can extend the notion of embeddedness in a pends on our ability to unambiguously at- cohesive group to the wider social network tribute social mechanisms to network fea- (Frank and Yasumoto 1998). The concept of tures. Connectivity provides researcherswith k-connected groups provides a clear the ability to disentangle the effects of struc- operationalization of a structural aspect of tural cohesion from other network features. embeddedness through the degree to which Using connectivity to capture a key di- actors' partners (or their partners' partners) mension of social cohesion is not a new idea, are connected to one another through multiple independent paths. As such, because 12 Similarly, an actor's nestedness in a cohe- cohesive groups are nested within one an- sive group is defined as the deepest outset level other, then each successive k-connected set within that group in which the actor resides. is more deeply embedded within the net- 13 D. White and Harary (2001) compare node work. This deep connectivity nicely captures connectivity approaches to many others, using data on Zachary's Karate Club as an exemplar. A the intuitive sense of being involved in rela- detailed comparison of each alternative method tions that are, in direct contrast to "arms- for measuring cohesion that expands on those D. length" relations, structurallyembedded in a White and Harary (2001), with multiple ex- social network (Uzzi 1996). As such, one as- amples, is available from the authors on request.

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS 113 though most previous approaches have fo- troduced in the literaturemore than 35 years cused on edge connectivity (Borgatti, ago, the ability to empirically employ these Everett, and Shirey 1990; Wasserman and ideas has only recently become available. Faust 1994). D. White and Harary (2001) Given the historical focus on small discuss the formal links between node and groups, is it reasonable to argue for "cohe- edge connectivity in detail. Briefly, a graph sion" in aggregates of many thousands of has edge connectivity k if it has no cutset of nodes? One might argue, for example, that a k- 1 edges, and, by Menger's Theorem, single loop connecting 1,000 people is not there are k edge-independent paths (as op- very cohesive. Why and when would such a posed to node-independent paths) connect- graph be considered cohesive? The answer, ing every pair of nodes in the graph. Al- as Markovsky and Lawler (1994) suggest, though the two concepts might seem intu- depends on the implicit comparison network. itively similar, they can result in radically Clearly, the substantive social character of a different assessments of group cohesion. 10-person group differs from that of a 1,000- Consider as an example Figure lb, which is person group. Comparison with a small pri- 2-edge-connected. By simply adding ties mary group will always give the impression, from node 7, one could increase edge con- if it has high density, that a large group with nectivity dramatically, but the graph as a lower density is less cohesive. We argue, whole would still depend entirely on node 7 however, that this is the wrong comparison to remain connected. As we discuss in more to make-that it conflates analytically dis- detail below, this kind of dominating central tinct dimensions of social structure, such as node would increase power inequality in the density or mean path distance, and the num- network and likely highlight divisions within ber of independent connections. Holding the the network. number of nodes and the density of a net- These substantive weaknesses in the edge work constant, the effect of greater node connectivity notion may explain why so few connectivity is always to increase social co- people have used it empirically, or have hesion. Structuralcohesion unites networks, found significant results with this method. independent of other factors such as size, Given the formal similarity between node with "independence"having the same mean- and edge connectivity, why hasn't node con- ing implied by most statistical models. Thus, nectivity been used before now? Although the correct comparison to make for a 1,000- many reasons are possible, including the person bicomponent is with a 1,000-person discipline's general focus on small primary spanning tree (less cohesive) or 1,000 people groups, the technical ability to identify high- divided into 250 four-person groups (less connectivity sets may be largely responsible cohesive yet). Other implicit comparisons, for its lack of use. Harary et al. (1965:25) of course, are various baseline models of were the first to propose node connectivity randomness. In a network of 1 million nodes as a measure of cohesion. A fast algorithm and 2 million edges, bicomponents in the to identify tricomponents was developed by range of 1,000 persons will be common, computer scientists in 1973 (Hopcroft and while a clique of 10 is an extremely rare Tarjan 1973), though it was never imple- event in a random network of 30 nodes and mented by social scientists, and the ability 50 edges. For a structurally cohesive group to identify bicomponents and pairwise node to be substantively significant within a net- connectivity is only now implemented in the work,14 whatever its number of nodes and most popular network software (Borgatti et al. 2002). The ability to identify the full connectivity of a graph as well as all cutsets is a 14We do not take up here the evaluation of sta- recent phenomenon, however. The algorithm tistical significance. D. White and Harary (2001) presented in Appendix A combines all the show that adding conditional density to node connectivity provides a continuous measure of necessary elements for a full cohesive block- connectivity that allows for comparability across ing, and in addition, provides a measure of networks of different sizes. An alternative ap- structurally cohesive embedding. Thus, proach to standardizing structural cohesion in- while the graph-theoretic ideas surrounding volves developing implicit comparison networks our approach to structuralcohesion were in- (also see Markovsky and Lawler 1994), such as a

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 114 AMERICAN SOCIOLOGICAL REVIEW number of edges, it must stand out against example illustrates how cohesive groups can the background of a relevant baseline model be identified in large settings based on of randomness. friendship, one of the most commonly stud- How does nestedness relate to other com- ied network relations. The second example mon network measures? To the extent that uses data on the interlocking directoratenet- nestedness captures the general location of works of 57 large firms in the United States actors and differentiates prominentactors, an (Mizruchi 1992). Because business solidar- actor's nestedness might be thought of as a ity has been an important topic of research type of centrality (Freeman 1977; Harary et on interlocks, we apply our method to this al. 1965; Wassermanand Faust 1994). How- network and show how structural cohesion ever, depth in the network is a group-level relates to similar political activity behaviors. property that distinguishes it from centrality Of course, there is not space here to treat the measures. Second, because connectivity is subtle theoretical issues surroundingeach of related to degree (each member of a k-com- these substantive areas. Instead, the analy- ponent must have at least k ties), nestedness ses below are designed to highlight how is necessarily correlated with degree. As we structuralcohesion can add to empirical re- show in the empirical examples below, how- search in widely differing research settings. ever, nestedness is not equivalent to any of these measures, either singly or in combina- STRUCTURAL COHESION IN ADOLESCENT tion, and measures something very different. FRIENDSHIP NETWORKS Although there are multiple dimensions upon which to compare a node connectivity Add Health is a school-based study of ado- concept of cohesion to alternatives, the real lescents in grades 7 through 12. A stratified test of the idea is whether it adds anything nationally representative sample of all pub- substantive to our understanding of empiri- lic and private high schools (defined as cal cases or gives rise to new theoretical hy- schools with an 11th grade) in the United potheses about social structure. Below we States with a minimum enrollment of 30 stu- demonstrate the empirical value-added of dents was drawn from the Quality Education our hierarchical concept of the relational Database (QED) in April 1994 (Bearman et component of solidarity in two radically dif- al. 1996). Network data were collected by ferent settings and then discuss further theo- providing each student with a copy of the retical implications of structuralcohesion. roster of all students for their school. Stu- dents identified up to 5 male and 5 female friends from this 16 TWO EMPIRICAL EXAMPLES: (10 total) roster. Here we HIGH SCHOOLS AND INTERLOCKING DIRECTORATES ment to the Carolina Population Center, Univer- sity of North Carolina at Chapel Hill, with coop- To demonstrate the empirical relevance of erative funding from 17 other agencies. Persons cohesive blocking, we use data from two dif- interested in obtaining data files from the Na- ferent types of networks. First, we use data tional Longitudinal Study of Adolescent Health on friendships among high school students should contact Add Health, Carolina Population Center, 123 West Franklin Street, taken from Chapel Hill, the National Longitudinal Study NC 27516-2524 ([email protected]). of Adolescent Health (Add Health).15 This 16 The maximum number of nominations al- lowed was 10, but this restriction affected few random network with similar degree distribution students. For all students, mean out-degree is or transitivity levels, which might be determined 4.15 with a standarddeviation of 3.02, 3 percent mathematically in simple cases (Newman, of students nominated 10 in-school friends, 23 Strogatz, and Watts 2001), or through Monte percent nominated 5 in-school male friends, and Carlo simulations as conditioning becomes more 25 percent named 5 in-school female friends. complex. Previous research suggests that close friendship 15 This research is based on data from the Add groups have about 5 members (Cotterell 1996; Health project, a program project designed by J. Dunphy 1963). Relations are likely to fall within Richard Udry (PI) and Peter Bearman, and gender. For the Add Health data as a whole (net funded by grant PO1-HD31921 from the National of other dyad attributes such as race, joint activi- Institute of Child Health and Human Develop- ties, and transitivity) same-sex ties are about 1.6

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS 115 use data on more than 4,000 students taken we use the mean score of the three items as from a dozen schools with between 200 and a measure of school attachment.Building on 500 students (mean = 349), providing a di- Markovsky and Lawler's (1994) discussion verse collection of public (83 percent) and of solidarity and cohesion, there ought to be private schools from across the United a significant positive relation between States. 17 nestedness in the network and school attach- NESTEDNESS AND SCHOOL ATTACH- ment, net of any other factor that might be MENT. For each school, we employed the associated with school attachment cohesive blocking procedure described in (Markovsky 1998; Markovsky and Chaffee Appendix A to identify all connectivity sets 1995; Markovsky and Lawler 1994; Paxton for each school friendship network. At the and Moody forthcoming). first level, we have the entire graph, which We control for other variables that might is usually unconnected (because of the pres- affect a student's school attachment. Be- ence of a small number of isolates). Most of cause gender differences in school perfor- the students in every school are contained mance and school climate are well known within the largest bicomponent, and often (Stockard and Mayberry 1992), we expect within the largest tricomponent. As the pro- female students to have lower school attach- cedure continues, smaller and more tightly ment than males. As students age we expect connected groups are identified. In these the school to become a less salient focus of data, at high levels of connectivity (k > 5), their activities, and grade in school is also subgroups do not overlap. This implies set- controlled.19 Students who perform well in tings with multiple cores, differentially em- school or who are involved in many extra- bedded in the overall school networks. When curricular activities should feel more com- no further cuts can be made within a group, fortable in schools. Because students from we have reached the end of the nesting struc- small schools might feel more attached than ture for that set of nodes. The level at which students from large schools, we test a this cutting ceases describes the nestedness school-level effect of size on mean school for each member of that group. attachment. Nestedness within the community should A significant feature of our approach is be reflected in a student's perception of his that we can differentiate the unique effects or her place in the school. The Add Health of network features that are often used to in-school survey asks students to report on measure "cohesion" in empirical work. First, how much they like their school, how close the number of contacts a person has (degree they feel to others in the school and how centrality) reflects their level of involvement much they feel a part of the school.18 Here, in the network. Substantively, we expect that those people with many friends in school are times more likely than cross-sex ties (Moody more likely to feel an integrated part of the 2001:712). For purposes of identifying connec- school. Second, having friends who are all tivity sets, we treat the graph as undirected. The friends with one another is an importantfea- algorithms needed for identifying connectivity ture of network involvement. As such, the can be modified to handle asymmetric ties. It was density of one's personal (local) network is for directed graphs that Harary et al. (1965) developed their concept of cohesion as connectiv- tested. Third, we expect that those people ity, although they offered no computational al- who are most central in the network should gorithms. have a greater sense of school attachment. 17 This sample represents all schools in the data Hence, betweenness centrality is tested. Fi- set of this size. The selected size provides a nice balance between computational complexity and ables are all positive, significant, and close in social complexity, as the schools are large magnitude. For similar work, see Resnick et al. enough to be socially differentiated and small (1997). The other three items used for Bollen and enough for group identification to be carried out Hoyle's scale were not included in the Add in a reasonable amount of computer time. Health school survey. 18 These are three items from the Perceived 19 Because school friendships tend to form Cohesion Scale (Bollen and Hoyle 1990), and within grade, controlling for grade in school cap- have a Chronbach's alpha reliability of .82. The tures an important focal feature of the in-school confirmatory factor loadings for the three vari- network.

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 11 6 AMERICAN SOCIOLOGICAL REVIEW nally, it may be the case that the lived com- through 6, we test the specification includ- munity of interest for any student is that set ing our measure and each of the four alter- of students with whom they interact most native network measures. In each case, often. We used NEGOPY (Richards 1995) to nestedness remains positive, significant, and identify density-based interaction groups strong, while inclusion of the alternative within the school, and use the relative group measures adds little explanatory power (as density to measure this effect.20 If our struc- seen by testing against Model 2). In Model tural aspect of nestedness captures a unique 7, we include all potentially confounding dimension of network embeddedness, as our network variables, and the relation between discussion above implies, then, controlling nestedness and attachment remains. The for each of these features, we would expect largest change in the coefficient for to find an independent effect of structurally nestedness comes with the addition of de- cohesive nestedness on school attachment.2" gree, which is likely due to collinearity, be- Table 1 presents HLM coefficients for cause every member of a k-component must models predicting school attachment from have degree ? k. nestedness level, school activity, demo- These findings suggest that individuals are graphic, and other network factors. Model 1 differentially attached to the school as a is a baseline model containing only attribute whole, and thus the school is differentially and school variables. As expected, females united, through structural cohesion. This and students in higher grades tend to have finding holds net of school-level differences lower school attachment,while students who in school attachment, the number of friends are involved in many extracurricularactivi- people have, the interaction densities among ties or who get good grades feel more at- their immediate friends or of their larger tached to the school. The coefficient for density-based interaction group, and their school size, while negative, is not statisti- betweenness centrality level. That these cally significant. In Model 2, our measure of other factors do not continue to contribute network nestedness is added to the model.22 additionally to school attachment implies a We see that there is a strong positive rela- unique effect of structural cohesion that tion between nestedness and school attach- other methods would wrongly have attrib- ment. (Note that the size of the standardized uted to the other measures of network struc- coefficient for nestedness is the largest in ture. this model.) Testing the difference in the de- COHESION AMONG LARGE AMERICAN viance scores between Model 1 and Model 2 BUSINESSES. A long-standing research tra- suggests that including nestedness improves dition has focused on the interlocking direc- the fit of the overall model. In Models 3 torates of large firms (Mizruchi 1982, 1992; Palmer, Friedlan, and Singh 1986; Roy 20 We thank an anonymous reviewer for sug- 1983; Roy and Bonacich 1988; Useem gesting this specification, which uses fewer de- 1984). An important question in this litera- grees of freedom than an alternative test that uses ture, "at the core of the debate over the ex- a dummy variable for each identified subgroup tent to which American society is demo- in the school. Tables with the alternative specifi- cratic" cation are available on request and show no sub- (Mizruchi 1992:32), is to what extent stantive difference in the nestedness effect. business in the United States is unified and, 21 We thank an anonymous reviewer for sug- if so, whether it is collusive. If businesses gesting a 2-level hierarchical linear model to test collude in the political sphere, then democ- for these relations, with students nested in racy is threatened.Yet much of the literature schools. The model was specified to allow coef- has been vague in defining exactly what con- ficients to vary randomly across schools, with the stitutes business unity, and thus empirical school-level intercept (substantively, mean determination of the extent and effect of school attachment) regressed on school size. business unity (and possible collusion) is 22 In addition to the nestedness level, we also hard to identify. tested a model using the largest k-connectivity value for each student. The results are very simi- Without treading on the issue of collusion lar. Students involved in high-cohesion groups per se, we approachthe question of business had higher levels of school attachment than oth- unity as a problem of structural cohesion. ers. Because structural cohesion facilitates the

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Table 1. Coefficients Predicting School Attachment from Network Embeddedness and Other Independent Variables: Add Health, 1994

IndependentVariable Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Intercept 3.850*** 3.429*** 3.361*** 3.467*** 3.354*** 3.322*** 3.379*** (.285) (.332) (.327) (.328) (.325) (.327) (.062) School size -.021 -.050 -.038 -.064 -.043 -.020 -.058 (.063) (.068) (.067) (.068) (.070) (.066) (.062) [-.015] [-.037] [-.027] [-.045] [-.030] [-.015] [-.041] Female -.189*** -.148*** -.155** -.149* -.153** -.149** -.161** (.04) (.001) (.041) (.042) (.042) (.043) (.044) [-.100] [-.075] [-.078] [-.076] [-.078] [-.076] [-.082] Gradein school -.077*** -.048* -.048* -.079* -.050* -.046 -.051 * (.018) (.022) (.021) (.021) (.021) (.022) (.022) [-.097] [-.059] [-.060] [-.060] [-.062] [-.059] [-.063] Grade-point .132*** .099*** .095** .102** .104*** .099** .099** average (.025) (.022) (.023) (.022) (.022) (.023) (.023) [.102] [.077] [.074] [.079] [.081] [.077] [.077] Extracurricular .115*** .076*** .078*** .076*** .075*** .076*** .076*** activities (.016) (.014) (.014) (.014) (.014) (.014) (.015) [.229] [.152] [.156] [.152] [.149] [.151] [.152] Nestedness .016*** .016*** .016*** .011*** .017*** .011i** (.001) (.001) (.001) (.002) (.001) (.003) [.279] [.277] [.268] [.201] [.283] [.181] Local density .002 .002 (.001) (.001) [.038] [.046] Betweenness .030 -.021 centrality (.027) (.043) [.023] [-.016] Number of friends .021* .029 (degree) (.010) (.014) [.087] [.118] Relative density .001 -.001 groups (.002) (.002) [.006] [-.014] Devianceb 9,842.31 9,577.57*** 9,578.29 9,581.13 9,571.06 9,587.60 9,574.31 Note: Numbersin parenthesesare standarderrors; numbers in bracketsare standardizedcoefficients. a School-level coefficient. b Numberof cases for level 1 = 3,606; numberof cases for level 2 = 12. *p < .05 **p< .01 ***p< .001 (two-tailedtests) flow of information and influence, coordi- large manufacturing firms, we identify the nated action, and thus political activity, ought cohesive group structure based on indirect to be more similar among pairs of firms that interlocks and relate this structure to simi- are similarly embedded in a structurallyco- larities in political action. hesive group. Mizruchi (1992) makes this The sample Mizruchi constructed consists argumentwell and highlights the importance of 57 of the largest manufacturing firms of financial institutions for unifying business drawn from "the twenty major manufactur- activity. He identifies the number of indirect ing industries in the U.S. Census Bureau's interlocks between two firms as "the number Standard Industrial Classification Scheme" of banks and insurance companies that have in 1980 (Mizruchi 1992:91). In addition to direct interlocks with both manufacturing data on directorship structure, he collected firms in the dyad" (p. 108). Using data on data on industry, common stockholding,

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 11 8 AMERICAN SOCIOLOGICAL REVIEW governmental regulations, and political ac- more financial stockholders two firms have tivity. The question of interest is whether the in common, the greaterthe similarity of their structureof relations among firms affects the political contributions.Additionally, indirect similarity of their behavior. To explore interlocks through financial institutions or whether firms that are similarly embedded jointly receiving defense contracts leads to also make similar political contributions, similarity of political action. In Model 2, we Mizruchi constructed a dyad-level political add the nestedness measure.25Net of the ef- contribution similarity score as a function of fects identified in Model 1, we find a strong the number of common campaign contribu- positive impact of cohesion within the indi- tions.23 He modeled this pair-level similar- rect interlock network. As in the school net- ity as a function of geographic proximity, works, we test for the potentially confound- industry, financial interdependence, govern- ing effects of degree and centrality.26No ef- ment regulations, and interlock structure. fect of network degree is evident, but be- A cohesive blocking of this network re- tweenness centrality does evidence a moder- veals that most firms are involved in a ate association with political similarity. strongly cohesive group, with 51 of the 57 When both variables are entered into the firms members of the largest bicomponent. model, the statistical significance of The nestedness structureconsists of a single nestedness drops slightly, but the magnitude hierarchy that is 19 layers deep, and at the of the effect remains constant. Based on the lowest level (at which no further minimum standardized coefficients, nestedness exhib- cuts can be made that would not isolate all its the strongest effect in each of the Models nodes), 28 firms are members of a 14-con- 2 through 5. The more deeply nested a given nected component (the strongest k-compo- dyad is in the overall network structure, the nent in the graph). more similar their political contributions. Does joint membership in more deeply The nestedness measure of structuralcohe- nested subsets lead to greater similarity in sion is a significant predictor of political political contributions? To answer this ques- similarity, in addition to the effect of direct tion, we add an indicator for the deepest adjacency created through financial inter- layer within which both firms in a dyad are locks. nested. Thus, if firm i is a member of the Mizruchi (1992) identifies two potential second layer but not the third, and firm j is a explanations for the importance of financial member of the fourth layer but not the fifth, interlocking on political behavior. Following the dyad is coded as being nested in the sec- Mintz and Schwartz (1985), banks and fi- ond layer. As with the school example, we nancial institutions may exercise control of control for other network features. Table 2 firms by seating representatives on their presents the results of this model. boards. As such, two firms that share many Model 1 replicates the analysis presented such financial ties face many of the same in- in Mizruchi (1992), and in the remaining fluencing pressures and therefore behave models we add additional network indica- similarly. A second argument, building on tors.24In the baseline model, we find that the the debate surrounding structural equivalence and cohesion (Burt 1978, 1982), is that 23 The score is calculated as actors in similar network positions (i.e., with similar of ties S. -ni patterns to similar third par- n. ties) ought to behave similarly. As in our argument for structural cohesion, Friedkin where Sij = the similarity score, nil equals the (1984) argues that influence travels through number of common campaign contributions, and ni and nj equal the number of contributions firm i 25 If instead of the joint nestedness level, we and j make, respectively. The dyad-level analy- use the connectivity level (k) for the highest k sis is based on 1,596 firm dyads. both members are involved in, we find substan- 24 Following Mizruchi (1992:121) we use the tively similar results. nonparametric quadratic assignment procedure 26 We cannot test for density-based subgroup (QAP) to assess the significance level of the re- effects because NEGOPY assigns all members to gression coefficients. See Mizruchi (1992) for the same group. This is a result of the high aver- measurement details. age degree within this network.

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Table 2. QAP Coefficients Predicting Political Action Similarity from Network Embeddedness and Other Dyad Attributes: 57 U.S. Manufacturing Firms, 1980

IndependentVariable VariableDescription Model 1 Model 2 Model 3 Model 4 Model 5 Proximity Headquarterslocated in same .017 .013 .013 .015 .015 state [.043] [.032] [.034] [.039] [.039] Same primary Same primarytwo-digit industry .012 .017 .017 .016 .016 industry [.024] [.034] [.036] [.034] [.034] Commonindustry Numberof common two-digit -.004 -.007 -.007 -.003 -.003 industries [-.008] [-.015] [-.015] [-.006] [-.006] Market constraint Interdependence based on .011 a .009 a .009 a .009 a .009 a transactionsand concentration [.098] [.080] [.082] [.083] [.083] Common stockholders Financialinstitutions that hold .034* .029* .028* .029* .029* stock in both firms [.213] [.182] [.174] [.181] [.181] Direct interlocks Board of directors overlaps .021 a .016 .017 a .018a .018a between firms [.047] [.036] [.037] [.041] [.041] Regulatedindustries Primarymembership in .036 a .034 a .032 .030 .030 regulatedindustry [.115] [.107] [.102] [.096] [.096] Defense contracts Commonrecipient of defense .084** .083** .082* .082* .082 contracts [.170] [.166] [.165] [.166] [.165] Indirectinterlocks Financialinstitutions with .026** .010** .009 .007 .007 which firms interlock [.178] [.070] [.060] [.050] [.051] Nestedness level Level of embeddednessin the .004* .005* .004* .004 a indirect interlocknetwork [.201] [.257] [.203] [.202] Degree difference Absolute difference in degree .001 -.00 [.087] [-.00] Centrality difference Absolute difference in between- .lO0a .010 ness centrality [.111] [.111] Constant .171** .156** .137** .144** .144** R-square .195 .217 .222 .229 .228 Source: Mizruchi(1992). Note: QuadraticAssignment Procedure(QAP) determinessignificance levels based on permutationtests and does not producenormal standarderrors. Numbers in bracketsare standardizedcoefficients. a Coefficient is significant at p < .10. We reportthis significance level in keeping with Mizruchi [1992]. *p < .05 **p< .01 (two-tailed tests) multiple paths and thus has an effect beyond to actor mechanisms (such as information the direct link between two actors. His argu- flow) to derive further theoretical conse- ment is supported by our finding that the quences of structural cohesion. A defining multiple, independent paths that link pairs of property of a k-component (by Menger's structurallycohesive actors help transfer in- Theorem) is that every pair of actors in the formation among firms in a way that is able collectivity is connected by at least k inde- to coordinate politically similar activity. pendent paths. The presence of multiple paths, passing through different actors, implies that if any one actor is removed, alter- THEORETICAL IMPLICATIONS OF native links among members remain to STRUCTURAL COHESION maintain social solidarity. Information and The above two empirical examples demon- resources can flow through multiple paths, strate the empirical validity of a structural making control of resources within the group conception of social cohesion. Because we by a small (< k) number of people difficult. have created a formal specification for struc- Although many potential implications likely tural cohesion, we can link network structure follow in particularsubstantive areas, we fo-

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 120 AMERICAN SOCIOLOGICAL REVIEW cus below on three broad types of sociologi- selves from the network), alternate paths cal questions: resource and risk flow, com- provide an opportunity for spread. munity and class formation, and power. Nonoverlapping (k + 1)-cohesive subgroups within a larger k-connected population have important implications for the RESOURCE AND RISK FLOW long-distance carrying capacity of the net- A focus on structuralcohesion provides new work. Local pockets of high connectivity act insights into diffusion, augmenting current as amplifying substations for information (or approachesthat focus largely on network dis- resource, or viral) flow that comes into the tance. The length of a path (numberof edges) more highly connected group, boosting a is often considered critical for the flow of signal's strength,30and sending it back out goods through a network, as flow may de- into the wider population. This pattern di- grade with relational distance. That is, the rectly reflects the core concept of sexually probabilitythat a resource flows between two transmitted diseases (Rothenberg et al. nonadjacent actors is equal to the product of 1996), which may account for the high each dyadic transition probability along the prevalence of many STDs in the face of path(s) connecting them. When multiplied quite low pair-wise transmission probabili- over long distances, the efficacy of the infor- ties. The observed patterns typical in small- mation diminishes, even if the pairwise trans- world graphs (Milgram 1969; Watts 1999) mission probability is high. For example, the are a natural result of local relational action probability that a message will arrive intact nested within a larger network setting. Thus, over a six-step chain27 when each dyadic processes based on the formal properties of transmission probability is .9 will be .53. The connectivity may account for many of the fragility of long-distance communication observed substantive features of small-world rests on the fact that at any step in the com- networks. munication chain, one person's failure to Social network researchershave tradition- pass the information will disrupt the flow. ally focused on small, highly connected For a structurally cohesive group, how- groups. Identifying connectivity as a central ever, expected information degradation de- element of cohesion frees us from focusing creases with each additional independent on small groups by identifying patterns path in the network. For example, the com- through which influence or information can parable probability of a six-step communi- travel long distances. The rise of electronic cation arriving given two independent paths communication and distributed information is .78.28As the number of independent paths systems suggest that distance will become increases, the likelihood of the information less salient as information can travel through transmission increases. When the flow is not channels that are robust to degradation. By subject to degradation, but only to interrup- extending our vision of cohesion from small tion, increasing connectivity will provide local groups to large extended relations, we faster and more reliable transmission are able to capture essential elements of throughout the network.29In a high-connec- large-scale social organization that have tivity network, even if many people stop only been hinted at by previous social net- transmission (effectively removing them- work research, providing an empirical tool for understanding realistically sized lived 27 This is the purported average acquaintance communities. distance among all people in the United States (Milgram 1969). 28 We calculate this as the product of the dy- COMMUNITY AND CLAss FORMATION adic probabilities for each path, minus the prob- Structural cohesion provides us with a use- ability of transmission through both paths. Thus, ful tool for understanding processes related for two paths of length d, the formula is 2(pj)d - (p1)2d, where d is the distance. This is a simplifi- cation, as dyadic transmission rates are often 30 Signal amplification might depend on aver- variable and highly context-specific. aging or combining degraded copies of the same 29 Computer viruses are an excellent example signal or message so as to filter noise, thus in- of such flows. creasing reliability.

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS 121 to the formation of social classes, ethnicity, friend's house while the friend similarly and social institutions. Although a long- claimed to be at theirs. In general, the emer- standing promise of network research gence of community through exchange oc- (Emirbayer 1997; Rapoport and Horvath curs when goods and information cycle 1961; H. White et al. 1976), the conceptual through the community, as evidenced clearly tools needed to identify the empirical traces in work on generalized exchange (Bearman of such processes have been sorely lacking. 1997). In contrast, Brudner and White (1997) showed that membership in a structurally POWER cohesive group based on marital and close kinship ties among households in an Aus- The substantive character of groups that are trian farming village was correlated with vulnerable to unilateral action differs signifi- stratified class membership, defined by cantly from that expected of groups with single-heir succession to ownership of the multiple independent connections. The productive resources of farmsteads and group as a whole is vulnerable to the will farmlands. and activities of those who can destroy the Linking structurally cohesive subgroup group by leaving. Moreover, actors that can membership to institutions that provide for- disconnect the group are also actors that can mal access to power suggests a new ap- control the flow of resources in the network. proach to the study of social stratification As has long been known from Network Ex- and the state. D. White et al. (2002), for exchange Theory, networks with structuralfea- ample, identify an informally organized "in- tures leading to control of resource flows visible state" created by the intersections of generate power inequality (Willer 1999). structurallycohesive groups across multiple In contrast to weak structurally cohesive administrative levels. They show that those groups, however, collectivities that do not who share administrative offices during depend on individual actors are less easily overlapping time spans build dense clique- segmented. The presence of multiple paths, like social ties within a political nucleus passing through different actors, implies that while maintaining sparse locally tree-like if any one actor is removed, alternative links ties with structurally cohesive groups (glo- among members still exist to maintain social bally multiconnected) in the larger region solidarity. Information and resources can and community. The locally dense and the flow through multiple paths, making minor- globally sparse multiconnected ties act as ity control of resources within the group dif- different kinds of amplifiers for the feedback ficult. As such, the inequality of power im- relations between larger cohesive groups and plicit in weakly cohesive structuresis not so their government representatives. pronounced in stronger structures. In gen- In his classic statement on the develop- eral, structurallycohesive networks are char- ment of social capital, Coleman (1988) ar- acterized by a reduction in the power pro- gued that a closed-loop structureconnecting vided by structuralholes (Burt 1982), as lo- adolescents' friends' parents increases effec- cal holes are closed at longer distances, unit- tive normative regulation in a community. ing the entire group. The key structural feature responsible for The development of "just-in-time"inven- this increased ability is that biconnected tory systems provides a compelling example. components (loops) allow information to When viewed as a network of resource flow freely throughout the community, al- flows, the most efficient production systems lowing normative ideas to be exchanged and resemble spanning trees, with tight cou- reinforced. Communities in which parents plings among plants. Under this structure, are connected to each other only indirectly labor has accentuated power because strikes, through adolescents will likely have weaker which effectively remove the struck factory normative regulation. Adolescents in such from the production network, disconnect the communities occupy a powerful position, entire production line. Recent trends toward controlling the flow of information.This fact "just-in-time" production processes are not is recognized by any teen that successfully new, but were used extensively early in the dupes parents into thinking they are at a auto industry. It became clear, however, that

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions 122 AMERICAN SOCIOLOGICAL REVIEW this production structure gave labor power. set to help provide clear-cut empirical stud- To counter, management expanded the pro- ies of embeddedness. duction network to include alternative Our analysis of structuralcohesion has fo- sources (other factories and storehouses), cused on the basic network features of so- building redundancy (i.e., structural cohe- cial cohesion, without regard to the particu- sion) into the system (Schwartz 2001). lar features that might be relevant in any given case. We suspect that researchers could modify aspects of our CONCLUSION AND DISCUSSION structural conception of cohesion as theory dictates. Thus, Social solidarity is a central concept in soci- in settings where flows degrade quickly, one ology. We have argued that solidarity can be could account for the level of cohesion by analytically divided into an ideational com- incorporating a measure of path length, tie ponent and a relational component (and per- strength, or the ratio of connectivity to group haps others). We have defined structural co- size. We caution, however, that much of the hesion as a measure of the relational compo- theoretical power of our concept of cohesion nent. The essential substantive feature of a rests on the idea that multiple indirect paths strongly cohesive group is that it has a sta- (perhaps routed through strongly cohesive tus beyond any individual group member. subgroups) can magnify signals such that We operationalize this concept of social co- long distances can be united through social hesion through the graph-theoretic property connections. Additionally, although we ex- of connectivity (Harary 1969; Harary et al. pect that structurally cohesive groups will 1965), showing that structural cohesion in- also be stable groups, this expectation must creases with each additional independent be tested empirically. path in a network. The qualitative relational feature we focus When does cohesion start? Following au- on, grounded in Simmel ([1908] 1950), is thors such as Markovsky and Lawler (1994), whether a group depends on particularindi- we argue that cohesion starts (weakly) when viduals for its group status. The relevant every actor can reach every other actor quantitative measure is the number of indi- through at least one relational path-the viduals whose involvement is required to paths that link actors are the social glue keep the group connected. Here we have ap- holding them together. We show that struc- plied our method in an effort to show how tural cohesion scales in that it is weakest cohesion might be profitably used in differ- when there is one path connecting actors, ent types of empirical settings. The settings stronger when there are two node-indepen- tested here are clearly only a small subset of dent paths, stronger yet with three node-in- the types of settings in which cohesion might dependent paths, and finally when, for n ac- be important, and our tests have focused on tors, there are almost as many (n - 1) inde- only one dimension of a decidedly complex pendent paths between each pair. concept. As with any single-dimensional Our conceptualization of structural cohe- network measure, our concept of structural sion simultaneously provides an operation- cohesion filters out a particularaspect of the alization of one of the structuraldimensions network. Previous literature has focused on of network embeddedness. Cohesive sets in alternativeelements, such as path distance or a network are nested, such that highly cohe- density. However, there are some aspects of sive groups are nested within less cohesive networks that might conceivably be impor- groups. Because the process for identifying tant for wider questions about solidarity that the nested connectivity sets is based on iden- are not captured in our measure. For ex- tifying the most fragile points in a network, ample, all currentmeasures treat networks as those actors who are involved in the most static, although the realization of any given highly connected portions of the network are network is time-dependent (Moody 2002). often deeply insulated from perturbationsin Future work might benefit by building time the overall network. Given the theoretical explicitly into models of social cohesion. importance of the generalized concept of Second, while our network focuses on the embeddedness in sociology, a measure of vulnerability of the network to node removal structuralembeddedness is an important as- (also see Borgatti 2002), we do not examine

This content downloaded from 128.235.251.160 on Mon, 2 Feb 2015 19:14:25 PM All use subject to JSTOR Terms and Conditions STRUCTURAL COHESION AND EMBEDDEDNESS 123 the probability that a given node will, in fact, James Moody is Assistant Professor of Sociol- be removed. For any given context, some ogy at The Ohio State University. His work fo- nodes may be more strongly entrenched in cuses on the formal properties of informal social the setting than others, which might provide relations, with particular interest in network dy- a contextual corollary to the ideas developed namics. Recent empirical work has examined the here. Finally, a direct link between the rela- social network basis of racial integration vol. 107, pp. tional structure and the ideational structure (American Journal of Sociology, 679-716) and dynamic constraints on network could be identified by layering observed so- diffusion (Social Forces, vol. 81, pp. 25-56). cial relations with ideational similarity mea- Currentprojects include a dynamic extension of sures, as are derived through shared mem- balance models, an NSF-funded project on email bership in groups or identification with par- relations, and collaboration on an NICHD- ticular ideas (Breiger 1974; Ennis 1992). funded study of HIV/STI risk networks. Furthertheory and research are requiredto understandhow relation type or strength af- Douglas R. White is Professor in the Institutefor fects the importance of cohesive structures Mathematical Behavioral Sciences and the An- for substantive outcomes. Of particular in- thropology Department at the University of Cali- terest will be work that, as with Durkheim's fornia-Irvine, where he also serves as Social Division of Labor ([1893] 1984), specifies Networks Graduate Program Director. He is an the relation between structuralcohesion and oft-visiting professor at the Ecole des Hautes ideational components of social solidarity Etudes in Paris, France, and member of several (Paxton and Moody forthcoming). The con- working groups at the Santa Fe Institute. He has published roughly 100 peer-reviewed articles, nection will require a sustained treatment of co-edited Research Methods in Social Network the ideational components of solidarity, as Analysis (George Mason Press, 1989) and Kin- might build from treatmentsof individual at- ship, Networks and Exchange (Cambridge Uni- tachment to a group (Bollen and Hoyle versity Press, 1998), and was recipient of the dis- 1990; Lawler 1992), or questions about tinguished von Humboldt Social Scientist Award identity (Hogg 1992). Our hope is that by in Germany. His recent research projects utilize providing a clear and concise definition and large-scale longitudinal network studies of com- operationalization of structural cohesion, munities and organizations to test newly opera- and a methodological tool for analysis, re- tional network theories of the consequences and searchers in many fields will be better able causes of social cohesion (for details see http:// to conduct their work. eclectic.ss. uci. edu/-d rwhite/nsf/w.htm).

APPENDIX A Cohesive Blocking Procedure for Identifying Connectivity Sets

Combiningalgorithms from computerscience (Ball identifythe component(Step 1), andidentify the cut- and Provan 1983; Even and Tarjan1975; Kanevsky node {7} (Step 2)." Separatingthe two subgraphsat 1990, 1993), we can identify cutsets in a networkas node 7 (Step 3) induces two new components: {7- follows: 16} and { 1-7,17-23 1 that are neither complete nor trivial. Within each induced subgraphwe repeatthe (1) Identify the connectivity, k, of the input graph. process, starting by identifying the subgraphcon- (2) Identify all k-cutsetsat the currentlevel of connectivity. Withinthe {7-16} bicomponent,we iden- nectivity. tify 18, 10), 110, 16), and 114, 16) as the 2-cuts for (3) Generate new graph components based on the this subgraph,each of which leads to a single mini- removal of these cutsets (nodes in the cutset be- mum degree cut-we call these types of cuts single- long to both sides of the inducedcut). ton cuts (e.g., of 9, 13, or 15. The graphremaining after the singleton cuts have been removed is t7, 8, (4) If the graph is neither complete nor trivial, re- 11, 14, 10, 12, and 16), which is 1-connected,with turnto 1; else end. t7, 8, 11, and 14 the largestincluded tricomponent). This procedureis repeated until all nested con- Because t7, 8, 11, and 14) form a completely connectivity sets have been enumerated.a Walking nected clique, we stop here and returnto the other through the example in Figure 2, we would first graphinduced by removing node 7, (t 1-7,17-23)).

a SAS IML programs for identifying the full connec- b An efficient algorithm for doing so can be found in tivity sets of a network are available (Moody 1999). Gibbons (1985).

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Again, this graph is a bicomponent.Cutsets {5, 7) ply these two procedures for every induced sub- and {21, 19), {21, 7}, and t5, 19) inducetwo graphs graph, and thus the total runningtime of the algo- of higher cohesion: {1-7) and {17-23} that are of rithmcan be substantial.Two steps can be taken to maximal connectivity, as further cuts will induce reduce the computationtime. First, there are linear- only singleton partitions. time algorithmsfor identifying k-connectedcompo- One can representthe hierarchicalnesting of con- nents for k < 3, and one can start searching with nectivity groups as a directed tree, with the total these algorithms, limiting the numberof levels at graph as the root, and each subgraphthat derives which one has to run the full connectivity algo- from it a new node. The cohesive blocking of a net- rithms (Fussell, Ramachandran,and Thurimella work consists of identifying all cohesive substruc- 1993; Hopcroftand Tarjan1973). Second, in many tures within the network and relating them to each empiricalnetworks the most common cutset occurs other in terms of the nested branchingof the sub- for singleton cuts. Because the procedureis nested, groups. The blocking for the example above is giv- one can search for nodes with degree less than or en in Figure 3 (with singleton cuts not represented). equal to the connectivity of the parent graph (the Testing for k-connectivity(Step 1) can be accom- graph from which the currentgraph was derived), plished with a network maximum flow algorithm remove them from the network, and thus apply the developed by Even and Tarjan (1975).c An algo- network flow search only after the singleton cuts rithmfor identifying all k-cutsetsof the graph(Step have been removed.e 2) was developed by Ball and Provan (1983), and was extended by Kanevsky (1990, 1993) to find all minimum-sizeseparating vertex sets.dOne must ap- e Additionally, there are approximation approaches (Auletta et al. 1999; Khuller and Raghavachari 1995) c This is an extension of Dinic's algorithm and runs that could be used to identify graph connectivity in O(V"12E2)time. within a certain amount of error, which would be d Which runs in 0(2kV3) time. faster.

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