Clustering coefficients for networks with higher order interactions

We introduce a clustering coefficient for nondirected and directed hypergraphs, which we call the quad clustering coefficient. We determine the average quad clustering coefficient and its distribution in real-world hypergraphs and compare its value with those of random hypergraphs drawn from the configuration model. We find that clustering in real-world hypergraphs is significantly different from those of random hypergraphs. Notably, we find that real-world hypergraphs exhibit a nonnegligible fraction of nodes with a maximal value of the quad clustering coefficient, while we do not find such nodes in random hypergraphs. Moreover, these highly clustered nodes are not observed in an analysis based on the pairwise clustering coefficient of the associated projected graph that has binary interactions, and hence higher order interactions are required to identify nodes with a large quad clustering coefficient.Comment: 29 pages, 18 figure