The critical percolation window in growing random graphs

TL;DR

This paper characterizes the critical percolation window in growing random graphs, revealing its width, phase transitions, and component sizes using coupling with branching random walks.

Contribution

It introduces a precise description of the critical window and phase transitions in a broad class of growing random graphs, including preferential attachment models.

Findings

Critical window width is of order (log n)^{-2}.

Largest component size inside the window is about √n / log n.

Susceptibility remains finite and constant across the window.

Abstract

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes.