How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse the microscopic mechanisms underlying discontin...

We propose a simulation model to study the properties of directed percolation in two-dimensional (2D) anisotropic random media. The degree of anisotropy in the model is given by the ratio μ between the axes of a semiellipse enclosing the bonds that promote percolation in one direction. At percolation, this simple model shows that the average number of bonds per site in 2D is an invariant equal ...

We report the critical point for site percolation for the “explosive” type for 2D square lattices using Monte Carlo simulations and compare it to the classical well known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the “explosive” site percolation threshold as pc = 0.695 and we find evidence that “explosive” site percolat...

Simultaneous capillary dominated displacement of the wetting and non-wetting phases are processes of interest in many disciplines including modeling of the penetration of polluting liquids in hydrology or the secondary migration in petroleum reservoir engineering. Percolation models and in particular invasion percolation is well suited to characterize the slow immiscible displaceme...

Site-bond percolation is a natural generalization of pure site percolation and pure bond percolation. The generalization allows both sites and bonds to be randomly occupied, with probabilities ps and pb respectively, in the case of random percolation. In particular, if pb = ps, site-bond percolation reduces to a pure site or pure bond percolation on a decorated lattice. It is also possible to i...

A hybrid model for soils, which combines percolation theory and finite element method is presented. The internal soil structure is modelled via the finite element method, and percolation networks are used for analyzing its mechanical behaviour. Through a microscopic characterization of elastic properties of soil grains, the model is generated. The effective percolation threshold obtained is low...

In this paper, we discuss the basic elements of percolation theory, beginning with the physical motivation for the abstraction and the essentials of the bond model on the LD square lattice. The critical phenomenon is introduced, a fundamental concept in percolation theory. We then describe the general use of percolation theory to analyze physical phenomena. We examine one such application in de...

We discuss inequalities and applications for percolation and randomcluster models. The relevant areas of methodology concern the following two types of inequality: inequalities involving the probability of a general increasing event, and certain differential inequalities involving the percolation probability. We summarise three areas of application of such inequalities, namely strict inequality...

Given ω ≥ 1, let Z2(ω) be the graph with vertex set Z 2 in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus Z2(1) is precisely Z .) Let pc(ω) be the critical probability for site percolation in Z2(ω). Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ ωpc(ω) = log(3/2). We also prove analogues of this resu...

Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena. Dynamical percolation is a model where a simple time dynamics is added to the (ordinary) percolation model. This dynamical model exhibits very interesting behavior. Our goal in this survey is to give an overview of the work in dynamical percolation that has been do...