Threshold Functions in Random s-Intersection Graphs

TL;DR

This paper studies threshold functions for key graph properties like perfect matchings and Hamilton cycles in random s-intersection graphs, showing they behave similarly to Erdős-Rényi graphs.

Contribution

It establishes threshold functions for various properties in binomial and uniform random s-intersection graphs, revealing their similarity to classical Erdős-Rényi graphs.

Findings

Threshold functions for perfect matching containment identified.

Threshold functions for Hamilton cycle containment established.

Results show similarity to Erdős-Rényi graph thresholds.

Abstract

Random $s$-intersection graphs have recently received considerable attention in a wide range of application areas. In such a graph, each vertex is equipped with a set of items in some random manner, and any two vertices establish an undirected edge in between if and only if they have at least $s$ common items. In particular, in a uniform random $s$-intersection graph, each vertex independently selects a fixed number of items uniformly at random from a common item pool, while in a binomial random $s$-intersection graph, each item in some item pool is independently attached to each vertex with the same probability. For binomial/uniform random $s$-intersection graphs, we establish threshold functions for perfect matching containment, Hamilton cycle containment, and $k$-robustness, where $k$-robustness is in the sense of Zhang and Sundaram [IEEE Conf. on Decision & Control '12]. We show that these threshold functions resemble those of classical Erd\H{o}s-R\'{e}nyi graphs, where each pair of vertices has an undirected edge independently with the same probability.