Related papers: Spread-out percolation in R^d
We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we…
We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…
In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a…
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…
Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric…
We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet.…
We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the…
We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\Bbb R}^d$. We present combinatorial expressions for the computation of the cumulants and…
The secrecy graph is a random geometric graph which is intended to model the connectivity of wireless networks under secrecy constraints. Directed edges in the graph are present whenever a node can talk to another node securely in the…
Equip each point $x$ of a homogeneous Poisson process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable…
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…
In the present article, numerical simulations have been performed to find the bond and site percolation thresholds on two-dimensional Gabriel graphs (GG) for Poisson point processes. GGs belong to the family of proximity graphs and are…
We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of…
We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a…
We consider first passage percolation on the Erd\H{o}s--R\'{e}nyi graph with $n$ vertices in which each pair of distinct vertices is connected independently by an edge with probability $\lambda/n$ for some $\lambda>1$. The edges of the…
A new random geometric graph model, the so-called secrecy graph, is introduced and studied. The graph represents a wireless network and includes only edges over which secure communication in the presence of eavesdroppers is possible. The…