Subcritical random hypergraphs, high-order components, and hypertrees

Abstract

One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph G(n, p), the threshold for the appearance of the unique largest component (also known as the giant component) is p g = n −1 . More precisely, when p changes from (1 − ε)p g (subcritical case) to p g and then to (1 + ε)p g (supercritical case) for ε > 0, with high probability the order of the largest component increases smoothly from O(ε −2 log(ε 3 n)) to Θ(n 2/3 ) and then to (1 ± o(1))2εn. Furthermore, in the supercritical case, with high probability the largest components except the giant component are trees of order O(ε −2 log(ε 3 n)), exhibiting a structural symmetry between the subcritical random graph and the graph obtained from the supercritical random graph by deleting its giant component.As a natural generalisation of random graphs and connectedness, we consider the binomial random k-uniform hypergraph H k (n, p) (where each k-tuple of vertices is present as a hyperedge with probability p independently) and the following notion of high-order connectedness. Given an integer 1 ≤ j ≤ k − 1, two sets of j vertices are called j-connected if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least j vertices. A j-connected component is a maximal collection of pairwise j-connected j-tuples of vertices. Recently, the threshold for the appearance of the giant j-connected component in H k (n, p) and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest j-connected components, with the help of a certain class of "hypertrees" and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant j-connected component.

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].…”

Hypergraphs for predicting essential genes using multiprotein complex data

“…Null models for hypergraphs have been developed in different domains. In uniform random hypergraphs all hyperedges have the same cardinality c [34,35]. In this study, we use two different null models (see Fig.…”