Evolution of high-order connected components in random hypergraphs

Abstract

We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the evolution of $j$-connected components in the $k$-uniform binomial random hypergraph $\mathcal{H}^k(n,p)$. In particular, we determine the asymptotic size of the giant component shortly after its emergence and establish the threshold at which the $\mathcal{H}^k(n,p)$ becomes $j$-connected with high probability. We also obtain a hitting time result for the related random hypergraph process $\{\mathcal{H}^k(n,M)\}_M$ -- the hypergraph becomes $j$-connected exactly at the moment when the last isolated $j$-set disappears. This generalises well-known results for graphs and vertex-connectivity in hypergraphs.Comment: Extended abstract presented at the European Conference on Combinatorics, Graph Theory and Applications 2015 summarising the results of arXiv:1501.07835 and arXiv:1502.07289, 6 page

Hypernetwork science via high-order hypergraph walks

“…In order to provide more a comprehensive approach to such questions, it would be of interest to determine conditions on model inputs under which certain s-walk properties of the output hypergraphs can be tightly bounded or controlled. While such work is outside the scope of the present paper, the aforementioned research by Kang, Cooley, and Koch [38][39][40] illustrates establishing guarantees on even basic high-order walk based properties in random hypergraphs (such as the size of the largest s-component) requires sophisticated probabilistic analysis.…”

“…Their motivation is to prove enumeration formulas for certain cycle structures in hypergraphs. In a series of three recent papers [38][39][40], Kang, Cooley, Koch, and others consider a notion of s-walk between s-tuples of vertices. They conduct a rigorous mathematical analysis of the asymptotic s-walk properties of binomial random k-uniform hypergraphs, considering hitting times, the evolution of high-order s-components, and high-order "hypertree" structures.…”

“…In comparison to their graph counterparts, generative hypergraph models are relatively few. While work on random uniform hypergraphs dates back to at least the 1970s, researchers have recently begun developing a wider variety of hypergraph models, both for uniform hypergraphs [38][39][40]67] and non-uniform hypergraphs [8,[68][69][70][71]. We consider three generative hypergraph models from [65], which can be thought of as hypergraph interpretations of the graph models Erdős-Rényi (ER) [72], Chung-Lu (CL) [73], and Block Two-Level Erdős-Rényi (BTER) [74,75].…”

Hypernetwork Science via High-Order Hypergraph Walks

“…Their motivation is to prove enumeration formulas for certain cycle structures in hypergraphs. In a series of three recent papers [15,16,17], Kang, Cooley, Koch, and others consider a notion of s-walk between s-tuples of vertices. They conduct a rigorous mathematical analysis of the asymptotic s-walk properties of binomial random k-uniform hypergraphs, considering hitting times, the evolution of highorder s-components, and high-order "hypertree" structures.…”

“…In comparison to their graph counterparts, generative hypergraph models are relatively few. Nonetheless, researchers have recently begun developing a wider variety of hypergraph models, both for the case of uniform hypergraphs [15,16,17,55] and non-uniform hypergraphs [11,19,20,26,35]. In the present work, we consider three generative hypergraph models from [2], which can be thought of as hypergraph interpretations of the graph models Erdős-Rényi (ER) [22], Chung-Lu (CL) [13], and Block Two-Level Erdős-Rényi (BTER) [41,63].…”