# Subhypergraphs in non-uniform random hypergraphs

## TL;DR

This paper investigates the occurrence of small subhypergraphs in large hypergraphs, highlighting limitations of reducing hypergraph problems to graph problems, and introduces a probabilistic model for analyzing subhypergraph expectations.

## Contribution

It introduces a generalized hypergraph random model and formalizes subhypergraph definitions, providing insights into their expected presence in random hypergraphs and comparing predictions with real data.

## Key findings

- Expected existence of subhypergraphs in random hypergraphs determined
- Model predictions align with real-world data on small subhypergraphs
- Clustering coefficients in hypergraphs are analyzed and compared

## Abstract

In this paper we focus on the problem of finding (small) subhypergraphs in a (large) hypergraph. We use this problem to illustrate that reducing hypergraph problems to graph problems by working with the 2-section is not always a reasonable approach. We begin by defining a generalization of the binomial random graph model to hypergraphs and formalizing several definitions of subhypergraph. The bulk of the paper focusses on determining the expected existence of these types of subhypergraph in random hypergraphs. We also touch on the problem of determining whether a given subgraph appearing in the 2-section is likely to have been induced by a certain subhypergraph in the hypergraph. To evaluate the model in relation to real-world data, we compare model prediction to two datasets with respect to (1) the existence of certain small subhypergraphs, and (2) a clustering coefficient.