Disorder-induced Limited Path Percolation

Abstract

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 αij 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 Erdős-Rényi 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.Percolation theory is the most well-established approaches to study system connectivity, and how this connectivity becomes compromised with local system failure [1]. Systems usually refer to connected structures such as lattices or random networks [2], and failures to the removal of nodes or links. Percolation predicts failure thresholds and size of the connected parts of the network after those failures. Practical domains of applicationes include epidemiology [3][4][5], communication networks like the Internet [6], and propagation of information in social networks [7].

Percolation on complex networks: Theory and application

“…In addition, if the communication of nodes in the network is effective only if the shortest path between nodes i and j after removal is shorter than αl i j , where l i j is the shortest path before removal, a much smaller failure of the network can lead to an effective network breakdown. This problem was also studied in a percolation form called limited path percolation [207,416]. Note that this mechanism was also used to feature the entanglement percolation in quantum complex networks [417].…”