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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 $tau in (2,3)$. Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as $ntoinfty$. 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 $tau in (3,4)$ and $tau >4$. Precisely, for edge-retention probabilities $pi_n = lambda n^{-(3-tau)/2}$, there is an explicitly computable $lambda_c>0$ such that the critical window is of the form $lambda in (0,lambda_c),$ where the largest clusters have size of order $n^{beta}$ with $beta=(tau^2-4tau+5)/[2(tau-1)]in[sqrt{2}-1, tfrac{1}{2})$ and have non-degenerate scaling limits, while in the supercritical regime $lambda > lambda_c$, a unique `tiny giant component of size $sqrt{n}$ emerges. For $lambda in (0,lambda_c),$ 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 $lambda>lambda_c$, we prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight $Omega(sqrt{n})$.
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variab
A bootstrap percolation process on a graph G is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected
A bootstrap percolation process on a graph $G$ is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infect
We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on th