Triadic Closure Research Articles

A puzzle concerning triads in social networks: Graph constraints and the triad census

Evidence from many sources shows that triadic tendencies are important structural features of social networks (e.g. transitivity or triadic closure) and triadic configurations are the basis for both theoretical claims and substantive outcomes (e.g. strength of weak ties, tie stability, or trust). A contrasting line of research demonstrates that triads in empirical social networks are well predicted by lower order graph features (density and dyads), accounting for around 90% of the variability in triad distributions when comparing different social networks (Faust, 2006, 2007, 2008). These two sets of results present a puzzle: how can substantial triadic tendencies occur when triads in empirical social networks are largely explained by lower order graph features? This paper provides insight into the puzzle by considering constraints that lower order graph features place on the triad census. Taking a comparative perspective, it shows that triad censuses from 159 social networks of diverse species and social relations are largely explained by their lower order graph features (the dyad census) through formal constraints that force triads to occur in narrow range of configurations. Nevertheless, within these constraints, a majority of networks exhibit significant triadic patterning by departing from expectation.

Origins of Homophily in an Evolving Social Network

The authors investigate the origins of homophily in a large university community, using network data in which interactions, attributes, and affiliations are all recorded over time. The analysis indicates that highly similar pairs do show greater than average propensity to form new ties; however, it also finds that tie formation is heavily biased by triadic closure and focal closure, which effectively constrain the opportunities among which individuals may select. In the case of triadic closure, moreover, selection to “friend of a friend” status is determined by an analogous combination of individual preference and structural proximity. The authors conclude that the dynamic interplay of choice homophily and induced homophily, compounded over many “generations” of biased selection of similar individuals to structurally proximate positions, can amplify even a modest preference for similar others, via a cumulative advantage–like process, to produce striking patterns of observed homophily.

Explaining the power-law degree distribution in a social commerce network

Social commerce is an emerging trend in which online shops create referral hyperlinks to other shops in the same online marketplace. We study the evolution of a social commerce network in a large online marketplace. Our dataset starts before the birth of the network (at which points shops were not linked to each other) and includes the birth of the network. The network under study exhibits a typical power-law degree distribution. We empirically compare a set of edge formation mechanisms (including preferential attachment and triadic closure) that may explain the emergence of this property. Our results suggest that the evolution of the network and the emergence of its power-law degree distribution are better explained by a network evolution mechanism that relies on vertex attributes that are not based on the structure of the network. Specifically, our analysis suggests that the power-law degree distribution emerges because shops prefer to connect to shops with more diverse assortments, and assortment diversity follows a power-law distribution. Shops with more diverse assortments are more attractive to link to because they are more likely to bring traffic from consumers browsing the WWW. Therefore, our results also imply that social commerce networks should not be studied in isolation, but rather in the context of the broader network in which they are embedded (the WWW).

Fundamental principles of network formation among preschool children

The goal of this research was to investigate the origins of social networks by examining the formation of children's peer relationships in 11 preschool classes throughout the school year. We investigated whether several fundamental processes of relationship formation were evident at this age, including reciprocity, popularity, and triadic closure effects. We expected these mechanisms to change in importance over time as the network crystallizes, allowing more complex structures to evolve from simpler ones in a process we refer to as structural cascading. We analyzed intensive longitudinal observational data of children's interactions using the SIENA actor-based model. We found evidence that reciprocity, popularity, and triadic closure all shaped the formation of preschool children's networks. The influence of reciprocity remained consistent, whereas popularity and triadic closure became increasingly important over the course of the school year. Interactions between age and endogenous network effects were non-significant, suggesting that these network formation processes were not moderated by age in this sample of young children. We discuss the implications of our longitudinal network approach and findings for the study of early network developmental processes.

Gender Clustering in Friendship Networks: Some Sociological Implications

This paper uses a social network approach to examine gender clustering in a complete network of teenagers and their friends. It demonstrates the advantages of using increasingly sophisticated social network techniques, including clustering coefficients and their visualization, and social selection models within the ERGM framework, to visualize and explain the process of clustering which takes place in teenagers' networks. The paper supports previous findings of gender homophily among teenagers in small cliques of friends, provides evidence of clustering among larger groups of friends that differs by gender and evidence that the process of clustering also differs by gender. Males make more friends and form larger clusters than females. Differences in clustering are due to differences in selection (males make more friends), triadic closure (more likely for females) and endogenous effects (impacting more on males). These findings have sociological implications for single-gender and cross-gender influences on teenagers' behaviour, and for the presumed importance of agency (selection) over structure (endogenous effects) on friendship formation.

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.

Wave Localization in Complex Networks with High Clustering

We show that strong clustering of links in complex networks, i.e., a high probability of triadic closure, can induce a localization-delocalization quantum phase transition (Anderson-like transition) of coherent excitations. For example, the propagation of light wave packets between two distant nodes of an optical network (composed of fibers and beam splitters) will be absent if the fraction of closed triangles exceeds a certain threshold. We suggest that such an experiment is feasible with current optics technology. We determine the corresponding phase diagram as a function of clustering coefficient and disorder for scale-free networks of different degree distributions P(k) approximately k;{-lambda}. Without disorder, we observe no phase transition for lambda<4, a quantum transition for lambda>4, and an additional distinct classical transition for lambda>4.5. Disorder reduces the critical clustering coefficient such that phase transitions occur for smaller lambda.

Explaining the Power-Law Degree Distribution in a Social Commerce Network

Social commerce is an emerging trend in which online shops create referral hyperlinks to other shops in the same online marketplace. We study the evolution of a social commerce network in a large online marketplace. Our dataset starts before the birth of the network (at which points shops were not linked to each other) and includes the birth of the network. The network under study exhibits a typical power-law degree distribution. We empirically compare a set of edge formation mechanisms (including preferential attachment and triadic closure) that may explain the emergence of this property. Our results suggest that the evolution of the network and the emergence of its power-law degree distribution are better explained by a network evolution mechanism that relies on vertex attributes that are not based on the structure of the network. Specifically, our analysis suggests that the power-law degree distribution emerges because shops prefer to connect to shops with more diverse assortments, and assortment diversity follows a power-law distribution. Shops with more diverse assortments are more attractive to link to because they are more likely to bring traffic from consumers browsing the WWW. Therefore, our results also imply that social commerce networks should not be studied in isolation, but rather in the context of the broader network in which they are embedded (the WWW).

Random networks with tunable degree distribution and clustering

We present an algorithm for generating random networks with arbitrary degree distribution and clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and Poisson degree distributions with variable levels of clustering. Such networks may be used as models of social networks and as a testable null hypothesis about network structure. Finally, we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network, and on the size of the giant component. Some analysis of these effects is presented.

Discussion

We demonstrate that future market correlation structure can be predicted with high out-of-sample accuracy using a multiplex network approach that combines information from social media and financial data. Market structure is measured by quantifying the co-movement of asset prices returns, while social structure is measured as the co-movement of social media opinion on those same assets. Predictions are obtained with a simple model that uses link persistence and link formation by triadic closure across both financial and social media layers. Results demonstrate that the proposed model can predict future market structure with up to a 40% out-of-sample performance improvement compared to a benchmark model that assumes a time-invariant financial correlation structure. Social media information leads to improved models for all settings tested, particularly in the long-term prediction of financial market structure. Surprisingly, financial market structure exhibited a higher predictability than social opinion structure.

The NP-Completeness of Edge-Coloring

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.