Threshold Functions in Random s-Intersection Graphs
Abstract
Random -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 common items. In particular, in a uniform random -intersection graph, each vertex independently selects a fixed number of items uniformly at random from a common item pool, while in a binomial random -intersection graph, each item in some item pool is independently attached to each vertex with the same probability. For binomial/uniform random -intersection graphs, we establish threshold functions for perfect matching containment, Hamilton cycle containment, and -robustness, where -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.
Cite
@article{arxiv.1502.00395,
title = {Threshold Functions in Random s-Intersection Graphs},
author = {Jun Zhao and Osman Yağan and Virgil Gligor},
journal= {arXiv preprint arXiv:1502.00395},
year = {2015}
}