On Thresholds for the Appearance of 2-cores in Mixed Hypergraphs

TL;DR

This paper investigates the thresholds for the emergence of 2-cores in mixed hypergraphs composed of different uniform hypergraphs, identifying configurations that maximize these thresholds to improve data structure efficiency.

Contribution

It provides a solution for optimal edge counts in mixed hypergraphs to maximize 2-core thresholds, surpassing those of uniform hypergraphs.

Findings

Maximized 2-core thresholds in mixed hypergraphs with specific edge sizes.

Demonstrated thresholds can exceed those of any single uniform hypergraph.

Implications for improving data structure space utilization.

Abstract

We study thresholds for the appearance of a 2-core in random hypergraphs that are a mixture of a constant number of random uniform hypergraphs each with a linear number of edges but with different edge sizes. For the case of two overlapping hypergraphs we give a solution for the optimal (expected) number of edges of each size such that the 2-core threshold for the resulting mixed hypergraph is maximized. We show that for adequate edge sizes this threshold exceeds the maximum 2-core threshold for any random uniform hypergraph, which can be used to improve the space utilization of several data structures that rely on this parameter.