Multiscale genesis of a tiny giant for percolation on scale-free random graphs
We study the critical behavior for percolation on inhomogeneous random networks on $n$ vertices, where the weights of the vertices follow a power-law distribution with exponent $τ\in (2,3)$. Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as $n\to\infty$. We identify the critical window for a host of scale-free random graph models such as the Norros-Reittu model, Chung-Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which, we see a sudden emergence of a tiny giant component. This is a novel behavior which is in contrast with the critical behavior in other known universality classes with $τ\in (3,4)$ and $τ>4$. Precisely, for edge-retention probabilities $π_n = λn^{-(3-τ)/2}$, there is an explicitly computable $λ_c>0$ such that the critical window is of the form $λ\in (0,λ_c),$ where the largest clusters have size of order $n^β$ with $β=(τ^2-4τ+5)/[2(τ-1)]\in[\sqrt{2}-1, \tfrac{1}{2})$ and have non-degenerate scaling limits, while in the supercritical regime $λ> λ_c$, a unique `tiny giant' component of size $\sqrt{n}$ emerges. For $λ\in (0,λ_c),$ the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on $\mathbb{Z}_+$ studied in a seminal work by Durrett and Kesten. For $λ>λ_c$, we prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight $Ω(\sqrt{n})$.