Competing bootstrap processes on the random graph $G(n,p)$

TL;DR

This paper studies two competing bootstrap percolation processes on large Erdős–Rényi graphs, analyzing how initial seeds and thresholds influence the final active red and black node sizes over time.

Contribution

It introduces a model with two competing activation processes on random graphs and characterizes their asymptotic behavior across different parameter regimes.

Findings

Final active sizes depend on initial seeds and thresholds.

Different regimes exhibit distinct asymptotic dynamics.

The model extends classical bootstrap percolation to competing processes.

Abstract

We extend classical bootstrap percolation by introducing two concurrent, competing processes on an Erd\H{o}s--R\'{e}nyi random graph $G(n,p_n)$. Each node can assume one of three states: red, black, or white. The process begins with $a_R^{(n)}$ randomly selected active red seeds and $a_B^{(n)}$ randomly selected active black seeds, while all other nodes start as white and inactive. White nodes activate according to independent Poisson clocks with rate 1. Upon activation, a white node evaluates its neighborhood: if its red (black) active neighbors exceed its black (red) active neighbors by at least a fixed threshold $r \geq 2$, the node permanently becomes red (black) and active. Model's key parameters are $r$ (fixed), $n$ (tending to $\infty$), $a_R^{(n)}$, $a_B^{(n)}$, and $p_n$. We investigate the final sizes of the active red ($A^{*(n)}_R$) and black ($A^{*(n)}_B$) node sets across different parameter regimes. For each regime, we determine the relevant time scale and provide detailed characterization of asymptotic dynamics of the two concurrent activation processes.