Disorder-induced Limited Path Percolation

¹ CABDyN Complexity Centre, Saïd Business School, University of Oxford - Park End Street, Oxford OX1 1HP, UK, EU
² Physics Department, Clarendon Laboratory, University of Oxford - Parks Road, Oxford OX1 3PU, UK, EU
³ Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Departamento de Física, Facultad de Ciencias Exactas y Naturales, UNMdP-CONICET - Funes 3350, (7600) Mar del Plata, Argentina
⁴ Center for Polymer Studies, Department of Physics, Boston University - 590 Commonwealth Ave., Boston, MA 02215, USA

Received: 21 November 2011
Accepted: 10 February 2012

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 the optimal path length 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.