Non-equilibrium coagulation processes and subcritical percolation on evolving networks

TL;DR

The paper analyzes percolation on growing uniform-attachment networks, revealing a distinct subcritical universality class with non-universal power-law scaling for component sizes. It develops a toolkit combining local weak limits, branching random walks, tree-graph inequalities, and stochastic-approximation to prove almost-sure scaling results: for π in (0,π_c) there exists α(π) in (0,1/2) with n^{−α(π)} times component sizes converging to positive limits, and the second susceptibility s_2^π(n) converges to a finite s_2^π(∞). It also shows the maximal component undergoes a BKT-like-like scaling indirectly through these dynamics, with strong evidence of long-range dependence and persistence of early-vertex influence; susceptibility remains bounded as π approaches π_c from below. The work introduces a general framework and tools transferable to broader evolving-network models, and outlines future directions on near-critical behavior and network-archaeology applications.

Abstract

We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical $π> π_c$ regime for a host of such models was conjectured in statistical physics, and then rigorously proven in mathematics, to exhibit behavior similar to the BKT infinite-order phase transition as $π\searrow π_c$. It has further been conjectured that the entire regime $π<π_c$ for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of $π\in (0, π_c)$. In this paper, we study percolation on the uniform attachment model, as a concrete template in order to develop general tools based on stochastic approximation, local convergence, branching random walks and tree-graph inequalities to prove the above conjectured phenomena. For each $π\in (0,π_c)$, we show there exists an explicit $α(π) \in (0,\tfrac{1}{2}) $ such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by $n^{α(π)}$, converge almost surely to strictly positive random variables as the network size $n \to \infty$. These dynamics lead to novel phenomena, compared to classical 'static' models, including long-range dependence and fixation of the identity of the maximal component, within finite time, among a finite number of 'early' components. Moreover, in contrast with most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and $π$ approaches $π_c$ from below. The general tools developed in this paper will be used in follow-up work to understand percolation for general growing network evolution models.

Figures (3)

• Figure 1.1: Density plots of $|\mathscr{C}^\pi_{\max}(n)|/n^{\alpha(\pi)}$ for two values of $\pi$. Each panel shows 200 trials per $n$, scaled by the predicted $n^{\alpha(\pi)}$, showing the non-constant nature of the limit.
• Figure 1.2: Both figures display the 6 largest components for $\pi=.14$ and the location of the first four vertices (in green) for $n=10^6$. In the left plot all of them are in the maximal component. In the right plot, 3 of the vertices appear in the second largest, and the fourth in the sixth largest, component.
• Figure 4.1: Example of the function $F$ for a specific choice of $\pi< \pi_c$.

Theorems & Definitions (54)

• Theorem 1.1: Convergence of fixed-vertex and maximal component sizes
• Remark 1: General $m$-out random graphs
• Remark 2: Novel universality class
• Theorem 1.2: Convergence of susceptibility
• Remark 3: Bounded critical susceptibility
• Definition 2.1: Multi-type branching random walk killed at a random barrier
• Remark 4: Moving the random barrier to zero
• Remark 5: Two-type killed branching random walk representation of general local limits
• Proposition 2.2: Local convergence of connected components
• proof