2025/12/17 by Banerjee, Sayan, Bhamidi, Shankar, van der Hofstad, Remco +1 · 2 citations
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Complex Network Analysis Techniques #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2512.15561
We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical π> πc 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 πc. It has further been conjectured that the entire regime π<πc for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of π∈ (0, πc). 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∈ (0,πc), we show there exists an explicit α(π) ∈ (0,\tfrac12) such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by nα(π), converge almost surely to strictly positive random variables as the network size n → ∞. 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 πc 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.