VERY LOCAL STRUCTURE IN SOCIAL NETWORKS

Triadic configurations are fundamental to many social structural processes and provide the basis for a variety of social network theories 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 82 social networks representing a number of different species (humans, baboons, macaques, bison, cattle, goats, sparrows, caribou, and more) and an assortment of social relations (friendship, negative sentiments, choice of work partners, advice seeking, reported social interactions, victories in agonistic encounters, dominance, and co-observation). Methodology uses low dimensional representations of triad censuses for these social networks, 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: network density, nodal indegree and outdegree distributions, and the dyad census. These findings reinforce the observation that structural configurations that can be realized in empirical social networks are severely constrained by very local network properties, making some configurations extremely improbable.     

Triadic configurations in limited choice sociometric networks: Empirical and theoretical results

Previous research demonstrated that information contained in triad censuses from heterogeneous collections of social networks occupies a high dimensional space. Regions of this space, and locations of triad censuses within it, are largely defined by lower order network properties: network density and dyad distributions [Faust, K., 2006. Comparing social networks: size, density, and local structure. Metodološki Zvezki (Advances in Methodology and Statistics) 3 (2), 185–216; Faust, K., 2007. Very local structure in social networks. Sociological Methodology 37, 209–256]. The current paper extends comparative work on triad censuses by addressing three related issues. First, it determines and interprets the space occupied by triad censuses for 128 friendship networks gathered using a limited choice sociometric protocol. Second, it constructs a theoretical space for triad censuses expected given lower order graph properties and examines the dimensionality and shape of this space. Third, it brings together these lines of investigation to determine where the empirical triad censuses reside within the theoretical space. Results show that the empirical triad censuses are almost perfectly represented in one dimension (explaining 99% of the data) and that network density explains over 96% of the variance in locations on this dimension. In contrast, the theoretical space for triad censuses is at least four-dimensional, with distinctive regions defined by network density and dyad distributions. Within this theoretical space, the empirical triad censuses occupy a restricted region that closely tracks triad censuses expected given network density. Results differ markedly from prior findings that the space occupied by triad censuses from heterogeneous social networks is of high dimensionality. Results also reinforce observations about constraints that network size and density place on graph level indices.     

Shared contexts and triadic closure in core discussion networks

This paper inquires into structural explanations for triadic closure in networks of confidants with whom one discusses important personal matters. Building upon the assumption that meeting opportunities affect network characteristics, we primarily argue that the social contexts in which network members meet, substantially affect triadic closure. The main empirical findings are (a) that about sixty percent of the triads in core discussion networks are closed triads, which also means that a substantial part of one's strong relations is unconnected, and (b) that meeting network members in the same social contexts is an important condition for, but certainly does not guarantee triadic closure. Importantly, the specific characteristics of social contexts explain why sharing certain contexts positively affects triadic closure, while sharing other contexts does not.     

Measuring social dynamics in a massive multiplayer online game

Quantification of human group-behavior has so far defied an empirical, falsifiable approach. This is due to tremendous difficulties in data acquisition of social systems. Massive multiplayer online games (MMOG) provide a fascinating new way of observing hundreds of thousands of simultaneously socially interacting individuals engaged in virtual economic activities. We have compiled a data set consisting of practically all actions of all players over a period of 3 years from a MMOG played by 300,000 people. This large-scale data set of a socio-economic unit contains all social and economic data from a single and coherent source. Players have to generate a virtual income through economic activities to ‘survive’ and are typically engaged in a multitude of social activities offered within the game. Our analysis of high-frequency log files focuses on three types of social networks, and tests a series of social-dynamics hypotheses. In particular we study the structure and dynamics of friend-, enemy- and communication networks. We find striking differences in topological structure between positive (friend) and negative (enemy) tie networks. All networks confirm the recently observed phenomenon of network densification. We propose two approximate social laws in communication networks, the first expressing betweenness centrality as the inverse square of the overlap, the second relating communication strength to the cube of the overlap. These empirical laws provide strong quantitative evidence for the Weak ties hypothesis of Granovetter. Further, the analysis of triad significance profiles validates well-established assertions from social balance theory. We find overrepresentation (underrepresentation) of complete (incomplete) triads in networks of positive ties, and vice versa for networks of negative ties. Empirical transition probabilities between triad classes provide evidence for triadic closure with extraordinarily high precision. For the first time we provide empirical results for large-scale networks of negative social ties. Whenever possible we compare our findings with data from non-virtual human groups and provide further evidence that online game communities serve as a valid model for a wide class of human societies. With this setup we demonstrate the feasibility for establishing a ‘socio-economic laboratory’ which allows to operate at levels of precision approaching those of the natural sciences.All data used in this study is fully anonymized; the authors have the written consent to publish from the legal department of the Medical University of Vienna.     

Testing transitivity in graphs

The situation is considered whether a graph can be assumed to have been generated by a random model capturing more transitivity than a simple uniform model. Three different test quantities based on induced triad counts and local densities are used. A simulation study is made in order to estimate critical values of the tests for different significance levels. The powers of the tests are estimated against the Bernoulli triangle model, a simple random graph model in which the clustering and transitivity is higher than in the uniform model. The test based on the proportion of transitive triads has the highest power in most cases, but the test based on density difference (the difference between mean local density and overall graph density) is more powerful against models with high transitivity. The tests are applied to a large set of school class sociograms. In this situation, uniform randomness is rejected in favor of transitivity most frequently when the test based on the proportion of transitive triads out of the non-vacuously transitive triads is used. It is concluded that this test, which also performed reasonably well when applied to random data, is the best at detecting transitivity. Although the Bernoulli triangle model fits to the empirical data set better than the uniform model, there are fewer truly intransitive triads in the data than could be expected under either of the models.     

Balance in stochastic signed graphs

Signed graphs provide models for investigating balance in connection with various kinds of social relations. Since empirical social networks always involve uncertainty because of errors due to measurement, imperfect observation or sampling, it is desirable to incorporate uncertainty into signed graph models. We introduce a stochastic signed graph and investigate the properties of some indices of balance involving triads. In particular we consider the balance properties of a graph which is randomly signed and of one which has been randomly sampled from a large population graph.     

Psychological predispositions and network structure: The relationship between individual predispositions,structural holes and network closure

We examine the effect of individual psychological differences on network structures, proposing several hypotheses about how individual differences might predispose actors to structure their social environment by seeking network closure or by sustaining structural holes. We introduce a new triad census method to examine personal networks of strong and weak ties. For 125 egocentric networks we correlated the triad census results with several extensively researched psychological instruments. The triad census reduced to three principal components, describing central aspects of strength-of-weak-ties and structural holes theories. Psychological predispositions explained a significant proportion of the variance in each of these components. Our results suggest that people who see themselves vulnerable to external forces tend to inhabit closed networks of weak connections. On the other hand, people who seek to keep their strong tie partners apart, and thereby bridge structural holes, tend to be individualists, to believe that they control the events in their lives, and to have higher levels of neuroticism. Finally, people with strong network closure and “weak” structural holes (as with the “strength of weak ties”) tend to categorize themselves and others in terms of group memberships. They also tend to be more extraverted and less individualistic.     

Informant accuracy in social network data III: A comparison of triadic structure in behavioral and cognitive data

This paper provides a comparison of the triadic-level structure inherent in behavioral and cognitive social network data taken on the same group, using a variety of groups whose communication could easily be monitored.It is found that many types of structure occur significantly more or less than chance in both behavioral and cognitive data, and providing that these are treated in similar ways, there is good agreement between the two structures. However, there are several ways to treat behavioral data, and these produce at least two essentially different structures.If cognitive and behavioral triads are compared, triad by triad, then there is virtually no agreement between them (even though they may both display the same structure on an overall triad census).Finally, as a demonstration of the dangers of relying solely on cognitive data, an unlikely null hypothesis is proposed. This asserts — for demonstration purposes — that, under many circumstances, behavioral structure never alters. Change in structure over time apparently occurs because of informant error in the reporting of the cognitive data. A pseudo-transition matrix, giving the probability that a triad is reported as one type when data are first taken, and a different type at a later date, is calculated. This compares reasonably with a genuine transition matrix evaluated for longitudinal cognitive data. It is believed that no data currently exist which can disprove this hypothesis, unlikely though it is. Much more accurate data are therefore necessary if any reliable theory of social structure is to be produced.     

Multiplexity and strategic alliances: The relational embeddedness of coalitions in social movement organisational fields

While scholars have embraced the notion of social movements as networks, there has been little empirical exploration of the emergence of coalitions within these multilayered systems. Here I explore the role of overlapping relations in alliance formation amongst a group of 55 health-related professional social movement organisations mobilised against austerity. Using cross-sectional bivariate exponential random graph models, I find dependencies between digital proxies for alliance, shared allies, information exchange, positive nomination and offline colobbying activity at the dyadic, degree and triadic levels. Cross-network associations indicate that multiplexity plays a non-trivial role in the formation of alliances and, more generally, social movement organisational fields, necessitating increased attention from scholars of social movements.     

Social context,spatial structure and social network structure

Frequently, social networks are studied in their own right with analyses devoid of contextual details. Yet contextual features – both social and spatial – can have impacts on the networks formed within them. This idea is explored with five empirical networks representing different contexts and the use of distinct modeling strategies. These strategies include network visualizations, QAP regression, exponential random graph models, blockmodeling and a combination of blockmodels with exponential random graph models within a single framework. We start with two empirical examples of networks inside organizations. The familiar Bank Wiring Room data show that the social organization (social context) and spatial arrangement of the room help account for the social relations formed there. The second example comes from a police academy where two designed arrangements, one social and one spatial, powerfully determine the relational social structures formed by recruits. The next example is an inter-organizational network that emerged as part of a response to a natural disaster where features of the improvised context helped account for the relations that formed between organizations participating in the search and rescue mission. We then consider an anthropological example of signed relations among sub-tribes in the New Guinea highlands where the physical geography is fixed. This is followed by a trading network off the Dalmatian coast where geography and physical conditions matter. Through these examples, we show that context matters by shaping the structure of networks that form and that a variety of network analytic tools can be mobilized to reveal how networks are shaped, in part, by social and spatial contexts. Implications for studying social networks are suggested.     

Relational collaboration among spatial multipoint competitors

The presence of network ties between multipoint competitors is frequently assumed but rarely examined directly. The outcomes of multipoint competition, therefore, are better understood than their underlying relational mechanisms. Using original fieldwork and data that we have collected on an interorganizational network of patient transfer relations within a regional community of hospitals, we report and interpret estimates of Exponential Random Graph Models (ERGM) that specify the probability of observing network ties between organizations as a function of the degree of their spatial multipoint contact. We find that hospitals competing more intensely for patients across multiple geographical segments of their market (spatial multipoint competitors) are significantly more likely to collaborate. This conclusion is robust to alternative explanations for the formation of network ties based on organizational size differences, resource complementarities, performance differentials, and capacity constraints. We show that interorganizational networks between spatial multipoint competitors are characterized by clear tendencies toward clustering and a global core-periphery structure arising as consequences of multiple mechanisms of triadic closure operating simultaneously. We conclude that the effects of competition on the structure of interorganizational fields depends on how markets as physical and social settings are connected by cross-cutting network ties between competitors.     

Triadic balance in the brain: Seeking brain evidence for Heider’s structural balance theory

Structural balance theory is a foundational theory of social network research. Despite enduring interest in seeking network evidence for the theory, the fundamental question of why people care about structural (im)balance has received relatively little attention. The original answer to the question, dating back to Heider’s work six decades ago, is that structural imbalance causes a person to experience cognitive dissonance and sentimental disturbance. In this paper, we used a state-of-the-art neuroimaging technique to test the argument. Our study shows that individuals’ psychological states, evidenced by the activation of brain areas, are different when they are situated in unbalanced rather than balanced triads. More specifically, the differences in the brain activation between triadic imbalance and balance were found in brain regions known for processing cognitive dissonance, as discovered by previous research. Our study provides novel brain evidence in support of Heider’s original account for the psychological and biological foundations of structural balance theory in the formation of social networks.     

Where do intra-organizational advice relations come from? The role of informal status and social capital in social exchange

Social status and social capital frameworks are used to derive competing hypotheses about the emergence and structure of advice relations in organizations. Although both approaches build on a social exchange framework, they differ in their behavioral micro-foundations. From a status perspective, advice giving is a means to generate prestige, whereas asking advice decreases one's relative standing. At a structural level these motivations are expected to result in an overrepresentation of non-reciprocal dyads and non-cyclical triadic structures in the advice network, as well as in active advice seekers being unlikely to be approached for advice, especially by active advice givers. From a social capital perspective, advice seeking creates obligations for the advice seeker. At the structural level, this results in an overrepresentation of reciprocal dyads and cyclical triads, and active advice seekers to be unpopular as targets of advice seeking, especially for active advice givers. Analyses of four waves of a longitudinal sociometric study of 57 employees of a Dutch Housing Corporation provide partial support for both approaches. In line with the social capital perspective, we find reciprocal advice relations to be overrepresented at the dyad level. Results at the triad level support the social status arguments, according to which high status individuals will avoid asking advice from low status individuals. The implications for macro-structural properties of intra-organizational advice network are discussed.     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time is the class of exponential random graph models (ERGMs), also known as p * models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, Markov chain Monte Carlo (MCMC) algorithms have been developed that produce approximate maximum likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data.
This paper proposes new specifications of exponential random graph models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistics are proposed: geometrically weighted degree distributions, alternating k -triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.     

Markov chain Monte Carlo exact inference for social networks

The uniformly most powerful unbiased test of reciprocity compares the observed number of mutual relations to its exact conditional distribution. Metropolis–Hastings algorithms have been proposed for generating from this distribution in order to perform Monte Carlo exact inference. Triad census statistics are often used to test for the presence of network group structure. We show how one of the proposed Metropolis–Hastings algorithms can be modified to generate from the conditional distribution of the triad census given the in-degrees, the out-degrees and the number of mutual dyads. We compare the results of this algorithm with those obtained by using various approximations.     

Assessing structural correlates to social capital in Facebook ego networks

Research in computer-mediated communication has consistently asserted that Facebook use is positively correlated with social capital. This research has drawn primarily on Williams’ (2006) bridging and bonding scales as well as behavioral attributes such as civic engagement. Yet, as social capital is inherently a structural construct, it is surprising that so little work has been done relating social capital to social structure as captured by social network site (SNS) Friendship networks. Facebook is particularly well-suited to support the examination of structure at the ego level since the networks articulated on Facebook tend to be large, dense, and indicative of many offline foci (e.g., coworkers, friends from high school). Assuming that each one of these foci only partially overlap, we initially present two hypotheses related to Facebook social networks and social capital: more foci are associated with perceptions of greater bridging social capital and more closure is associated with greater bonding social capital. Using a study of 235 employees at a Midwestern American university, we test these hypotheses alongside self-reported measures of activity on the site. Our results only partially confirm these hypotheses. In particular, using a widely used measure of closure (transitivity) we observe a strong and persistent negative relationship to bonding social capital. Although this finding is initially counter-intuitive it is easily explained by considering the topology of Facebook personal networks: networks with primarily closed triads tend to be networks with tightly bound foci (such as everyone from high school knowing each other) and few connections between foci. Networks with primarily open triads signify many crosscutting friendships across foci. Therefore, bonding social capital appears to be less tied to local clustering than to global cohesion.     

What makes a network go round? Exploring the structure of a strong component with exponential random graph models

A strong component is a subgraph in a directed network where, following the direction of ties, all nodes in the graph are reachable from one another. Mutual reachability implies that every node in the graph is theoretically able to send materials to and/or influence every other node suggesting that strong components are amongst the more egalitarian network structures. Despite this intriguing feature, they remain understudied. Using exponential random graph models (ERGM) for directed networks, we investigate the social and structural processes underlying the generation of strong components. We illustrate our argument using a network of 301 nodes and 703 personal lending ties from Renaissance Florence. ERGM shows that our strong component arises from triadic clustering alongside an absence of higher-order star structures. We contend that these processes produce a strong component with a hierarchical, rather than an egalitarian structure: while some nodes are deeply embedded in a dense network of exchange, the involvement of others is more tenuous. More generally, we argue that such tiered core-periphery strong components will predominate in networks where the social context creates conditions for an absence of preferential attachment alongside the presence of localized closure. Although disparate social processes can give rise to hierarchical strong components linked to these two structural mechanisms, in Florence they are associated with the presence of multiple dimensions of social status and the connectedness of participants across disparate network domains.     

Closure,connectivity and degree distributions: Exponential random graph (p*) models for directed social networks

The new higher order specifications for exponential random graph models introduced by Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology 36, 99–153] exhibit substantial improvements in model fit compared with the commonly used Markov random graph models. Snijders et al., however, concentrated on non-directed graphs, with only limited extensions to directed graphs. In particular, they presented a transitive closure parameter based on path shortening. In this paper, we explain the theoretical and empirical advantages in generalizing to additional closure effects. We propose three new triadic-based parameters to represent different versions of triadic closure: cyclic effects; transitivity based on shared choices of partners; and transitivity based on shared popularity. We interpret the last two effects as forms of structural homophily, where ties emerge because nodes share a form of localized structural equivalence. We show that, for some datasets, the path shortening parameter is insufficient for practical modeling, whereas the structural homophily parameters can produce useful models with distinctive interpretations. We also introduce corresponding lower order effects for multiple two-path connectivity. We show by example that the in- and out-degree distributions may be better modeled when star-based parameters are supplemented with parameters for the number of isolated nodes, sources (nodes with zero in-degrees) and sinks (nodes with zero out-degrees). Inclusion of a Markov mixed star parameter may also help model the correlation between in- and out-degrees. We select some 50 graph features to be investigated in goodness of fit diagnostics, covering a variety of important network properties including density, reciprocity, geodesic distributions, degree distributions, and various forms of closure. As empirical illustrations, we develop models for two sets of organizational network data: a trust network within a training group, and a work difficulty network within a government instrumentality.     

Social context and network formation: An experimental study

Recently, there has been increasing interest in determining which social network structures emerge as a consequence of the conscious actions of actors. Motivated by the belief that “networks matter” in reaching personal objectives, it is a natural assumption that actors try to optimize their network position. Starting from the notion that an optimal network position depends on the social context, we examine how actors change their networks to reach better positions in various contexts. Distinguishing between three social contexts (a neutral context, a context in which closed triads are costly, and a context in which closed triads are beneficial), theoretical results predict that emerging networks are contingent on the incentives that are present in these contexts. Experiments are used to test whether networks that are theoretically predicted to be stable are also stable experimentally. We find that emerging networks correspond to a large extent with the predicted networks. Consequently, they are contingent on the incentives present in various social contexts. In addition, we find that subjects tend to form specific stable networks with a higher probability than predicted, namely, efficient networks and networks in which everyone is equally well off.