Weighted transitivity scores that account for triadic edge similarity in undirected 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. When the connections vary in strength, focusing on whether connections 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. 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 in males compared to females, and consistent with the triadic process underlying goose dominance relationships.