Transitivity scores to account for triadic edge weight similarity in undirected weighted graphs

The graph transitivity measures the probability that adjacent vertices in a network are interconnected, thus revealing the existence of tightly connected neighborhoods playing a role in information and pathogen circulation. The graph transitivity is usually computed for dichotomized networks, therefore focusing on whether triangular relationships are closed or open. But when the connections vary in strength, focusing on whether the closing ties exist or not can be reductive. I score the weighted transitivity according to the similarity between the weights of the three possible links in each triad. In a simulation, that new technique correctly diagnosed excesses of balanced or imbalanced triangles, for example, strong triplets closed by weak links. I illustrate the biological relevance of that information with two reanalyses of animal contact networks. In the rhesus macaque Macaca mulatta, a species in which kin relationships strongly predict social relationships, the new metrics revealed striking similarities in the configuration of grooming networks in captive and free-ranging groups, but only as long as the matrilines were preserved. In the barnacle goose Branta leucopsis, in an experiment designed to test the long-term effect of the goslings' social environment, the new metrics uncovered an excess of weak triplets closed by strong links, particularly pronounced in males, and consistent with the triadic process underlying goose dominance relationships.