A random triadic process

TL;DR

This paper investigates a process on random hypergraphs where edges are added based on existing triangles, establishing a threshold probability for the process to propagate and implications for the connectivity of random simplicial complexes.

Contribution

It introduces a new probabilistic process on hypergraphs, determining the threshold for propagation and its connection to the simple connectivity of random complexes.

Findings

Threshold probability for propagation is 1/(2√n).

Propagation occurs with high probability above this threshold.

Implication for simple connectivity of random 2-complexes.

Abstract

Given a random 3-uniform hypergraph $H=H(n,p)$ on $n$ vertices where each triple independently appears with probability $p$, consider the following graph process. We start with the star $G_0$ on the same vertex set, containing all the edges incident to some vertex $v_0$, and repeatedly add an edge $xy$ if there is a vertex $z$ such that $xz$ and $zy$ are already in the graph and $xzy \in H$. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is $p=\frac{1}{2\sqrt{n}}$. We conclude that $p=\frac{1}{2\sqrt{n}}$ is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.