Counting simplicial pairs in hypergraphs

TL;DR

This paper introduces measures to quantify the simplicial interactions in hypergraphs, analyzes real-world data, and proposes a new model to simulate such interactions, revealing different behaviors in stochastic processes.

Contribution

It presents the simplicial ratio and matrix as new tools for analyzing hypergraph structure and introduces a Chung-Lu model incorporating simplicial interaction control.

Findings

Simplicial interactions tend to increase with edge size in real-world hypergraphs.

The new Chung-Lu model can simulate varying levels of simplicial interactions.

Different stochastic processes behave distinctly on simplicial versus non-simplicial hypergraphs.

Abstract

We present two ways to measure the simplicial nature of a hypergraph: the simplicial ratio and the simplicial matrix. We show that the simplicial ratio captures the frequency, as well as the rarity, of simplicial interactions in a hypergraph while the simplicial matrix provides more fine-grained details. We then compute the simplicial ratio, as well as the simplicial matrix, for 10 real-world hypergraphs and, from the data collected, hypothesize that simplicial interactions are more and more deliberate as edge size increases. We then present a new Chung-Lu model that includes a parameter controlling (in expectation) the frequency of simplicial interactions. We use this new model, as well as the real-world hypergraphs, to show that multiple stochastic processes exhibit different behaviour when performed on simplicial hypergraphs vs. non-simplicial hypergraphs.