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Register a new userMultiscale genesis of a tiny giant for percolation on scale-free random graphs
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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 $Omega(sqrt{n})$.
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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 variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passing times.
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 and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0.
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 infected and remains so forever. The parameter $rgeq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $beta$, where $2 < beta < 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)gg a_c(n)$, then there is a constant $eps>0$ such that, with high probability, the final set of infected vertices has size at least $eps n$. It turns out that when the maximum degree is $o(n^{1/(beta -1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Theta (n^{1/(beta -1)})$, then $a_c (n)=n^{beta -2 over beta -1}$.
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 uninfected vertex becomes type 1 (2) infected at rate $lambda_1$ ($lambda_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $lambda_1=lambda_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $Vin(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $lambda_1 eq lambda_2$, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.
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 the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.
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