A Study of Percolation Phenomena in Process Networks

Abstract. Percolation theory provides models for a wide variety of natural phenom-ena. One of these phenomena is the dielectric breakdown of composite materials.This paper describes how we implemented the percolation model for dielectric break-down in a massively parallel processing environment. To achieve this we modifiedthe breadth-first search algorithm such that it works in probabilistic process networks.Formalmethodswereusedtoreasonaboutthisalgorithm.Furthermore,thisalgorithmprovides the basis for a JCSP implementation which models dielectric breakdownsin composite materials. The implementation model shows that it is possible to applyformal methods in probabilistic processing environments.Keywords. Percolation, Breadth-first search, JCSP, Probabilistic process networks,channel poisoning IntroductionThepercolationproblemhasbeenextensivelyinvestigatedbymathematiciansandphysicists.In this paper we analyse the problem from a computer science point of view. One of thereasons for this cross disciplinary interest is the fact that the percolation problem yields ex-tremelysimplesystemsexhibitingtheintriguingcomplexitiesofphasetransitions.Thesesys-temscanbeassociatedwithmanyphysicalrealisations[1,2].S.R.BroadbentandJ.M.Ham-mersley posed the original percolation problem in the context of graph theory [3]. They con-sidered an arbitrary linear graph, like the one shown in Figure 1, in which the vertices arepoints of the model, and a given pair of vertices is linked by an edge with probability pindependently of all other pairs; this is an example of bond percolation.