VERY LOCAL STRUCTURE in SOCIAL NETWORKS Katherine

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VERY LOCAL STRUCTURE in SOCIAL NETWORKS Katherine 7 VERY LOCAL STRUCTURE IN SOCIAL NETWORKS Katherine Faust* Triadic configurations are fundamental to many social structural processes and provide the basis for a variety of social network the- ories and methodologies. This paper addresses the question of how much of the patterning of triads is accounted for by lower-order properties pertaining to nodes and dyads. The empirical base is a collection of 82social networks representing a number of dif- ferent species (humans, baboons, macaques, bison, cattle, goats, sparrows, caribou, and more) and an assortment of social rela- tions (friendship, negative sentiments, choice of work partners, ad- vice seeking, reported social interactions, victories in agonistic en- counters, dominance, and co-observation). Methodology uses low dimensional representations of triad censuses for these social net- works, as compared to censuses expected given four lower-order social network properties. Results show that triadic structure is largely accounted for by properties more local than triads: net- work density, nodal indegree and outdegree distributions, and the dyad census. These findings reinforce the observation that struc- tural configurations that can be realized in empirical social net- works are severely constrained by very local network properties, making some configurations extremely improbable. I am grateful to John Boyd, Carter Butts, and Kim Romney for insightful discussions of this research and to members of the UCI social network research group for providing a forum to present these ideas. Two reviewers and the editor of Sociological Methodology provided veryhelpful comments on an earlier draft. I thank Lin Freeman for sharing data on a number of nonhuman social networks. Direct correspondence to Katherine Faust at [email protected]. *Universityof California, Irvine 209 210 FAUST 1. BACKGROUND Network approaches to social structure view human and nonhuman an- imal populations as systems of relations among interacting units—with examples including communications, exchange of material resources, expressions of affection, informal social interactions, kinship relations, victories in agonistic encounters, exercise of authority, and provision of social support. Relations among triads—triples of units—are funda- mental to structural patterns in manyof these relations and considerable theoretical, empirical, and methodological work in the social sciences concerns triads. Simmel (1950) is often credited with the earlyinsight that social processes are fundamentallydifferent when three rather than two people are involved, thus focusing attention on the triad. The third person in an interaction affords possibilityfor mediation between the other two, the formation of coalitions, and advantages accruing through tertius strategies such as brokerage. Appreciation of the importance of triadic processes has been carried forward in a number of ways, and several in- fluential social theories rest, at least in part, on triadic patterns. Theories on cognitive balance (Heider 1946) and its generalization to structural balance (Cartwright and Harary1956) focus on the strain involved in triads when sentiments are not consonant—for example when two close friends are in strong disagreement about their evaluation of another per- son or object. Granovetter’s (1973) classic strength of weak ties argu- ment proceeds from the strong (admittedlyoverstated) assumption that when an individual, A, has strong ties with two others, B and C, then the tie between B and C should not be absent. A “forbidden triad” is one that violates this assumption. Granovetter then follows the social struc- tural implications of the absence of forbidden triads for the occurrence of bridging weak ties, social integration, and information diffusion in social networks. Related ideas are found in Burt’s (1992) structural holes argument and its extensions. The argument is that efficiencyand effec- tiveness result from nonredundant ties. A person who has ties to others who are not themselves tied bridges a structural hole. This pattern, which often forms an open triangle, is an important locus for strategic action. Triadic processes are also implicated in third-partyeffects on trust and reputation (Burt and Knez 1995). Third parties can affect dyadic trans- actions, in which theyare not directlyinvolved, byconveying(or being in a position to convey) information about the actions of others, thus VERY LOCAL STRUCTURE 211 influencing their reputations. Third-partyeffects are also important in interactions among nonhuman animals, and theyhave been most exten- sivelyinvestigated in dominance encounters (Chase 1982; Chase, Tovey, and Murch 2003). Empirical evidence demonstrates that the presence of third parties affects the instabilityof dominance orderings and the persistence of disadvantage from prior losses. Moreover, a triadic “by- stander” effect might be necessaryfor development of dominance hi- erarchies (Skvoretz, Faust, and Fararo 1996). Though in verydifferent contexts from Simmel’s (1950) original theoretical reasoning, these em- pirical results similarlychallenge an assumption that valid conclusions about dyadic interactions can be reached by studying them in isolation. These lines of research point to triadic configurations and in- teractions among triples as important for larger social processes. Since triads concern relations among social actors, most often theyare stud- ied using a social network formalization. Indeed, triads have been at the heart of theoretical and methodological advances in social network research for nearlyhalf a century.Triads are implicated in manysocial network theories, as noted above. Triads and properties of triples also provide the basis for important social network methods—for example, the triad census (Holland and Leinhardt 1970), random and biased nets (Skvoretz, Fararo, and Agneessens 2004), the formal linkage between lo- cal network processes and global social structures (Johnsen 1998), tran- sitivityindices (Frank and Harary1982), triad based role equivalence (Burt 1990), and structural effects in manyexponential random graph models (Kalish and Robins 2006; Robins and Pattison 2005; Robins et al. forthcoming; Snijders et al. 2006). The importance of triads in social structural investigation can hardlybe overstated, as Holland and Leinhardt argue, The essential issue of anynotion of structure is how the components are combined, not the components themselves...this issue amounts to the proposition that the lowest interesting level of structure...is the level of triples of nodes—the triadic level (1979:66). Holland and Leinhardt then recognize that nodal and dyadic properties constrain possible triadic structures that might be realized empirically, and they ask whether social network data contain any 212 FAUST information beyond that expected from lower-level properties: con- straints imposed bydata collection methods (e.g., question format), differences in popularityamong actors, and a tendencyfor sociometric choices to be reciprocated (Holland and Leinhardt 1979). The extended investigations of triads byHolland and Leinhardt as well as others fol- lowing their lead tackled these related questions, but at no point have theyanswered the question: How much of the triadic structure is ac- counted for bynodal and dyadicproperties? This paper provides an answer to that question and shows that more than 90% of the triadic structure in a collection of diverse social networks is accounted for by lower-order properties. The implications of this result are far-reaching. If a substantial portion of triadic structure is explained bynodal and dyadic features, then the theoretical and methodological importance of triadic properties and processes are brought into question. The analyses that follow extend the work of Skvoretz and Faust on social network comparisons (Faust and Skvoretz 2002; Skvoretz and Faust 2002) and especiallythe results of Faust (2006), demonstrating that triad censuses for a collection of 51 social networks are largely explained bylinear and quadratic functions of network densityand dyad distributions. This paper uses a different sample of 82 social net- works and a different methodological approach, comparing triad cen- suses for these observed networks with censuses expected given four different lower-order network properties: (1) network density, (2) the outdegree distribution, (3) the indegree distribution, and (4) the dyad census. The analysis employs low-dimensional representations of ob- served and expected triad censuses followed bycanonical redundancy analysis to quantify the exact percentage of variance in the observed triad censuses that is explained bytriad censuses expected given the lower-order network properties. 2. LOCAL STRUCTURE IN SOCIAL NETWORKS A social network, represented as a graph or directed graph, consists, minimally, of a set of nodes (also referred to as points or vertices) rep- resenting social actors and a set of arcs (edges or ties) between pairs of nodes, representing social relations between actors. A graph with node set V and arc set E is denoted G(V, E). A social network with g actors can be displayed in a g × g sociomatrix, X, in which rows and columns VERY LOCAL STRUCTURE 213 index actors, in identical order, and entries, xij, code the state of the arc from actor i to actor j. This minimal social network maybe elaborated byallowing valued arcs, multiple relations, arcs that are nondyadic(i.e., that include more than two actors), have more than two sets
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