Clustering coefficients for networks with higher order interactions

TL;DR

This paper introduces a new clustering coefficient for hypergraphs, called the quad clustering coefficient, and demonstrates its effectiveness in distinguishing real-world hypergraph structures from random models, revealing highly clustered nodes with large degrees and hyperedges.

Contribution

The paper proposes the quad clustering coefficient for hypergraphs and analyzes its distribution in real-world versus random hypergraphs, highlighting the importance of higher-order interactions.

Findings

Real-world hypergraphs have nodes with maximal quad clustering coefficient.

Highly clustered nodes can have large degrees and hyperedges.

The quad clustering coefficient captures higher-order interactions not seen in pairwise analysis.

Abstract

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 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. Interestingly, these highly clustered nodes can have large degrees and can be incident to hyperedges of large cardinality. Moreover, 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.