A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs

Abstract

A bootstrap percolation process on a graph with infection threshold r ≥ 1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever.Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Z d that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson, Luczak, Turova and Vallier (2012). In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model : each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.

High-dimensional bootstrap processes in evolving simplicial complexes

“…This critical behaviour was also proved for the classic (graph) bootstrap process on the Barabási-Albert model by the first author and Abdullah [1] as well as by Ebrahimi et al [16]. In a different context, such a phenomenon was also shown in inhomogeneous random graphs [3,18] as well as in random graphs on the hyperbolic plane [12]. In [18] it is proved that in the class of inhomogeneous random graphs (Chung-Lu model) such a transition occurs essentially only when the degree distribution follows a power law with exponent less than 3.…”

“…In a different context, such a phenomenon was also shown in inhomogeneous random graphs [3,18] as well as in random graphs on the hyperbolic plane [12]. In [18] it is proved that in the class of inhomogeneous random graphs (Chung-Lu model) such a transition occurs essentially only when the degree distribution follows a power law with exponent less than 3.…”

Bootstrap percolation on geometric inhomogeneous random graphs

“…where the last estimate holds by (36). Therefore, by Theorem 17 each vertex in T d \ 2 i1+1 B i0−1 of weight at most log log n is in V ≤i0+i1 with probability at most…”

“…It was originally developed to model various physical phenomena (see [1] for a short review), but has by now also become an established model for the spreading of activity in networks, for example for the spreading of beliefs [31,40,58,61], behaviour [38,39], or viral marketing [50] in social networks (see also [22]), of contagion in economic networks [7], of failures in physical networks of infrastructure [65] or compute architecture [35,51], of action potentials in neuronal networks (e.g, [6,24,32,33,56,60,62,63], see also [53] for a review), and of infections in populations [31]. Bootstrap percolation has been intensively studied theoretically and experimentally on a multitude of models, including trees [10], lattices [3,9], Erdős-Rényi graphs [48], various geometric graphs [15,37,55], and scale-free networks [8,12,29,36,50]. On geometric scale-free networks there are some experimental results [21], but little is known theoretically.…”

Bootstrap percolation in random geometric graphs

“…Bootstrap percolation has been rigorously studied on several other random graph models: random regular graphs [10,26], power-law random graphs [4], Bienaymé-Galton-Watson trees [11], random graphs with specified vertex degrees [26], toroidal grids with random long edges added in (a special case of the Kleinberg model) [27], inhomogeneous random graphs [20], amongst others. A motivation for the latter two models is that they may have degree distributions or spatial characteristics that more closely resemble those of the real-life networks motivating the study of bootstrap percolation.…”