Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is ...

Graphene flakes with giant shape anisotropy are extensively used to establish connectedness electrical percolation in various heterogeneous systems. However, the percolation behaviour of graphene flakes has been recently predicted to be far more complicated than generally anticipated on the basis of excluded volume arguments. Here we confirm experimentally that graphene flakes self-assemble int...

A number of centrality measures are available to determine the relative importance of a node in a complex network, and betweenness is prominent among them. However, the existing centrality measures are not adequate in network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of dis...

Häggström, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree, Peres and Steif (1998) derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The ...

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 threshold. We model the street systems as PoissonVoronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequat...

Let 0 < a < b < ∞, and for each edge e of Zd let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d > 1 the distance distω(0, v) from the origin to a vertex v, |v| > 2, has variance bounded by C |v|/ log |v|, where C = C(a, b, d) is a constant which may only depend on a, b a...

In this article we study percolation on the Cayley graph of a free product of groups. Such a graph has a tree-like structure which allows us to evaluate the critical values of the phase transition, mean cluster size and the critical exponent in bond percolation.

This letter investigates the multiple routes transmitted epidemic process on multiplex networks. We propose detailed theoretical analysis that allows us to accurately calculate the epidemic threshold and outbreak size. It is found that the epidemic can spread across the multiplex network even if all the network layers are well below their respective epidemic thresholds. Strong positive degree-d...