First, we will briefly introduce oriented percolation with our focus on a special orientation on the square grid. Then we will discuss critical probabilities of bond or site percolation on this particular orientation. Finally, we will introduce the Fire-fighting Problem and prove a theorem regarding an upper bound for these critical probabilities in the case that fractional fire-fighters were a...

We analytically solve the core percolation problem for complex networks with arbitrary degree distributions. We find that purely scale-free networks have no core for any degree exponents. We show that for undirected networks if core percolation occurs then it is continuous while for directed networks it is discontinuous (and hybrid) if the in- and out-degree distributions differ. We also find t...

In bridge percolation one gives a special weight to bridges, i.e., bonds that if occupied would create the first spanning cluster. We show that, for p > pc, even far away from the critical point of classical percolation, the set of bridge bonds is fractal with a fractal dimension dBB = 1.215±0.002. This new percolation exponent is related to various different models like, e.g., the optimal path...

Percolation of particle arrays is of high interest in microstructural design of materials. While there have been numerous contributions to theoretical modeling of percolation in particulate systems, no analytical approximation for the generalized problem of variable aspect-ratio ellipses has been reported. In the present work, we (1) derive, and verify through simulation, an analytical percolat...

Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or almost completely unaffected based on the initial seed being larger or smaller than the critical value. In this paper, we consider behavioral interventions,...

The studied linear discrete-continuous system contains two coupled subsystems: one with continuous-time dynamics, the other with discrete-time dynamics. Continuoustime dynamics are described by ordinary linear differential equations, whereas discretetime dynamics are described by difference equations for the system state jumps at prescribed time instants. Systems of three following degrees of g...

Erdös and Rényi proved in 1960 that a drastic change occurs in a large random graph when the number of edges is half the number of nodes: a giant connected component surges. This was the origin of percolation theory, where statistic mechanics and mean field techniques are applied to study the behavior of graphs and other structures when we remove edges randomly. In the 90’s the study of random ...

In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density one, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area Ac for percolation in this model. The proof is in two parts: first we give a rigorous reduction of the problem to a finite problem, an...

The September 11 attack on the US has revealed an unprecedented terrorism worldwide range of destruction. Recently, it has been related to the percolation of worldwide spread passive supporters. This scheme puts the suppression of the percolation effect as the major strategic issue in the fight against terrorism. Accordingly the world density of passive supporters should be reduced below the pe...