Sampling uniform hypergraphs with given degrees

Graphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple k-uniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a black-box bipartite graph sampler. This algorithm was presented in a paper by myself and co-authors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple k-uniform hypergraphs with degree sequence d. Our algorithm uses a black-box sampler A for producing (possibly non-simple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is near-regular. This extends the range of degree sequences for which efficient sampling schemes are known