High-order bootstrap percolation in hypergraphs

TL;DR

This paper extends bootstrap percolation to hypergraphs, analyzing the minimal initial infection size needed for complete percolation in various hypergraph configurations.

Contribution

It introduces a high-order generalization of bootstrap percolation in hypergraphs and determines the exact minimal percolating set size for most cases.

Findings

Exact minimal initial infected set size in hypergraphs

Generalization of bootstrap percolation to hypergraphs

Results applicable to various hypergraph parameters

Abstract

Motivated by the bootstrap percolation process for graphs, we define a new, high-order generalisation to $k$-uniform hypergraphs, in which we infect $j$-sets of vertices for some integer $1\le j \le k-1$. We investigate the smallest possible size of an initially infected set which ultimately percolates and determine the exact size in almost all cases of $k$ and $j$.