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

TL;DR

The paper investigates two competing bootstrap-like activation processes on the Erdős--Rényi random graph $G(n,p)$ with three node states: red, black, and white. Activation occurs asynchronously via Poisson clocks and follows a threshold rule: a white node becomes red (or black) if its red (or black) active neighbors exceed those of the opposite color by at least $r\ge2$. The authors establish a multi-scale fluid-limit framework: on time-scale $q$ the dynamics converge to a deterministic Cauchy problem, which then governs the stopping time $K^*$ and final active sizes $A_R^*$ and $A_B^*$; in subcritical regimes, both colors grow to finite limits, while in supercritical regimes one color (typically red) percolates to a linear or full graph size, drastically constraining the other color. They provide explicit results for the canonical case $r=2$, derive closed-form expressions for key thresholds like $\kappa_{oldsymbol g}$ and $g_B(\kappa_{oldsymbol g})$, and develop a rigorous analysis that extends to time-scales beyond $q$, showing rapid domination by the advantaged color. The methodology combines stochastic process thinning, Markovian dynamics, and a Cauchy-ODE framework to characterize asymptotic trajectories, with numerical illustrations confirming the theoretical predictions. Overall, the work advances bootstrap percolation theory by handling competition between two irreversible activation processes on random graphs and reveals how time-scale and seed scaling dictate global outcomes with potential implications for information spread and competing cascades on networks.

Abstract

We extend classical bootstrap percolation by introducing two concurrent, competing processes on an Erdős--Rényi 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.

Figures (4)

• Figure 1: Case $q=g$, $\alpha_R > 1 > \alpha_B$: $A_B^*/q$ as function of $\alpha_R$, for different values of $\alpha_B$.
• Figure 2: Case $q=g$, $\alpha_R > \alpha_B > 1$: $A_B^*/q$ as function of $\alpha_R$, for different values of $\alpha_B$.
• Figure 3: Logic dependencies among scale $q$ results; $A\to B$ means that $A$ is invoked in the proof of $B$.
• Figure 4: Logic dependencies among scale $q$ results; $A\to B$ means that $A$ is invoked in the proof of $B$.

Theorems & Definitions (62)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7: sub-critical regime
• Theorem 2.8: super-critical regime
• Definition 2.9
• Remark 2.10