# Higher-order clustering in networks

## TL;DR

This paper introduces higher-order clustering coefficients to measure the likelihood of closure in larger network cliques, extending traditional clustering concepts to better understand complex network structures.

## Contribution

It proposes a natural generalization of the clustering coefficient for higher-order cliques and analyzes their properties and applications in real-world networks.

## Key findings

- Higher-order clustering coefficients provide new insights into network structure.
- They generalize traditional clustering and reveal complex clustering patterns.
- Application to real-world networks shows their practical utility.

## Abstract

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect to such higher-order network structures is not well understood. Here we introduce higher-order clustering coefficients that measure the closure probability of higher-order network cliques and provide a more comprehensive view of how the edges of complex networks cluster. Our higher-order clustering coefficients are a natural generalization of the traditional clustering coefficient. We derive several properties about higher-order clustering coefficients and analyze them under common random graph models. Finally, we use higher-order clustering coefficients to gain new insights into the structure of real-world networks from several domains.