Linear algebra and bootstrap percolation

TL;DR

This paper investigates the minimum initial infected set needed for complete spread in hypergraph bootstrap percolation, using linear algebra techniques to analyze powers of complete graphs and induced subgraph structures.

Contribution

It introduces a novel linear algebra approach to bootstrap percolation, specifically for hypergraphs encoding induced copies of a graph in powers of complete graphs.

Findings

Determines minimum initial infected set size for complete infection.

Applies linear algebra techniques to bootstrap percolation problems.

Connects bootstrap percolation with weakly saturated graphs.

Abstract

In $\HH$-bootstrap percolation, a set $A \subset V(\HH)$ of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph $\HH$. A particular case of this is the $H$-bootstrap process, in which $\HH$ encodes copies of $H$ in a graph $G$. We find the minimum size of a set $A$ that leads to complete infection when $G$ and $H$ are powers of complete graphs and $\HH$ encodes induced copies of $H$ in $G$. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent to (edge) $H$-bootstrap percolation on a complete graph.