Current Redistribution in Resistor Networks: Fat-Tail Statistics in Regular and Small-World Networks

TL;DR

This paper studies how electrical currents redistribute after bond failures in resistor networks, revealing fat-tail statistics that depend on network topology and resistances, with implications for understanding failure propagation.

Contribution

It introduces a framework to analyze current redistribution factors based solely on network topology and bond resistances, highlighting fat-tail behavior in regular and small-world networks.

Findings

Current redistribution factors depend only on network topology and bond resistances.

Fat-tail statistics indicate long-range effects in current redistribution.

Long-range redistribution is influenced by network structure and Kirchhoff's laws.

Abstract

The redistribution of electrical currents in resistor networks after single-bond failures is analyzed in terms of current-redistribution factors that are shown to depend only on the topology of the network and on the values of the bond resistances. We investigate the properties of these current-redistribution factors for regular network topologies (e.g. $d$-dimensional hypercubic lattices) as well as for small-world networks. In particular, we find that the statistics of the current redistribution factors exhibits a fat-tail behavior, which reflects the long-range nature of the current redistribution as determined by Kirchhoff's circuit laws.