Related papers: Phase transitions and percolation at criticality i…
We study the percolation phase transition on preferential attachment models, in which vertices enter with $m$ edges and attach proportionally to their degree plus $\delta$. We identify the critical percolation threshold as…
We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK…
We define an inhomogeneous percolation model on "ladder graphs" obtained as direct products of an arbitrary graph $G = (V,E)$ and the set of integers $\mathbb{Z}$ (vertices are thought of as having a "vertical" component indexed by an…
We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood…
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…
We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p)…
We consider random graphs $\mathcal{G}$ built on a homogeneous Poisson point process on $\mathbb{R}^d$, $d\geq 2$, with points $x$ marked by i.i.d. random variables $E_x$. Fixed a symmetric function $h(\cdot, \cdot)$, the vertexes of…
The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive…
In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge…
A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the…
This thesis considers three models which describe a multihop ad-hoc telecommunication system. These systems consist of users sending messages, which can jump to other users to reach the target user. The first two models have already been…
In this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model.…
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…
In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…
We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville…
Cascading failures in complex systems have been studied extensively using two different models: $k$-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically and validate our…
The percolated random geometric graph $G_n(\lambda, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a…
A necessary and sufficient condition is established for the strict inequality $p_c(G_*)<p_c(G)$ between the critical probabilities of site percolation on a quasi-transitive, plane graph $G$ and on its matching graph $G_*$. It is assumed…
We study inhomogeneous Bernoulli bond percolation on the graph $G \times \mathbb{Z}$, where $G$ is a connected quasi-transitive graph. The inhomogeneity is introduced through a random region $R$ around the origin axis…
We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on $\mathbb{R}^d$. This class includes the age-dependent random connection model and the soft Boolean…