Clustering function: another view on clustering coefficient

Assuming that actors u and v have r common neighbors in a social network we are interested in how likely is that u and v are adjacent. This question is addressed by studying the collection of conditional probabilities, denoted cl(r), r = 0, 1, 2, . . . , that two randomly chosen actors of the social network are adjacent, given that they have r common neighbors. The function r → cl(r) describes clustering properties of the network and extends the global clustering coefficient. Our empirical study shows that the function r → cl(r) exhibits a typical sigmoid pattern. In order to better understand this pattern we establish the large scale asymptotics of cl(·) for two related random intersection graph models of affiliation networks admitting a non-vanishing global clustering coefficient. key words: Clustering coefficient, social network, affiliation network, clustering function, random intersection graph.