📝 SummarXiv

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.