Bootstrap Percolation on Degenerate Graphs

TL;DR

This paper studies bootstrap percolation on degenerate graphs, providing bounds on the final infected set size based on initial conditions and graph degeneracy, with implications for understanding infection spread in sparse networks.

Contribution

It introduces a new upper bound on the final infected set size for bootstrap percolation on d-degenerate graphs when r>d, advancing theoretical understanding.

Findings

Final infected set size is bounded by (1+d/(r-d)) times the initial infected set size.

The bound applies specifically to finite d-degenerate graphs with r>d.

Provides a theoretical framework for infection spread limits in sparse graph classes.

Abstract

In this paper we focus on $r$-neighbor bootstrap percolation, which is a process on a graph where initially a set $A_0$ of vertices gets infected. Now subsequently, an uninfected vertex becomes infected if it is adjacent to at least $r$ infected vertices. Call $A_f$ the set of vertices that is infected after the process stops. More formally set $A_t:=A_{t-1}\cup \{v\in V: |N(v)\cap A_{t-1}|\geq r\}$, where $N(v)$ is the neighborhood of $v$. Then $A_f=\bigcup_{t>0} A_t$. We deal with finite graphs only and denote by $n$ the number of vertices. We are mainly interested in the size of the final set $A_f$. We present a theorem for degenerate graphs that bounds the size of the final infected set. More precisely for a $d$-degenerate graph, if $r>d$, we bound the size set $A_f$ from above by $(1+\tfrac{d}{r-d})|A_0|$.