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.