Disorder Induced Limited Path Percolation

Abstract

We introduce a model of percolation induced by disorder, where an initially homogeneous network with links of equal weight is disordered by the introduction of heterogeneous weights for the links. We consider a pair of nodes i and j to be mutually reachable when the ratio α_{ij} of length of the optimal path between them before and after the introduction of disorder does not increase beyond a tolerance ratio τ. These conditions reflect practical limitations of reachability better than the usual percolation model, which entirely disregards path length when defining connectivity and, therefore, communication. We find that this model leads to a first order phase transition in both 2-dimensional lattices and in Erdos-Renyi networks, and in the case of the latter, the size of the discontinuity implies that the transition is effectively catastrophic, with almost all system pairs undergoing the change from reachable to unreachable. Using the theory of optimal path lengths under disorder, we are able to predict the percolation threshold. For real networks subject to changes while in operation, this model should perform better in predicting functional limits than current percolation models.