Sampling hypergraphs with given degrees

TL;DR

This paper presents a rejection sampling method for generating simple hypergraphs with specified degrees by leveraging bipartite graph sampling algorithms, analyzing conditions for efficiency and success probability.

Contribution

It introduces a novel approach connecting hypergraph sampling to bipartite graph algorithms, providing conditions for efficient and reliable sampling.

Findings

The algorithm's expected runtime depends on bipartite graph sampling efficiency.

Conditions are identified under which the sampling probability remains bounded away from zero.

The method enables uniform sampling of hypergraphs with given degree sequences.

Abstract

There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm $\mathcal{A}$, and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a positive constant.