Ising model for distribution networks

TL;DR

This paper introduces an Ising spin model to simulate cascading failures in distribution networks, analyzing how network topology and policies influence the likelihood of collapses.

Contribution

It develops a novel Ising model with quenched random fields to study failure mechanisms in distribution networks, combining numerical simulations and mean-field analysis.

Findings

Hubs play a critical role in initiating network failures.

Effective temperature influences the likelihood of cascading collapses.

Network topology affects susceptibility to breakdowns.

Abstract

An elementary Ising spin model is proposed for demonstrating cascading failures (break-downs, blackouts, collapses, avalanches, ...) that can occur in realistic networks for distribution and delivery by suppliers to consumers. A ferromagnetic Hamiltonian with quenched random fields results from policies that maximize the gap between demand and delivery. Such policies can arise in a competitive market where firms artificially create new demand, or in a solidary environment where too high a demand cannot reasonably be met. Network failure in the context of a policy of solidarity is possible when an initially active state becomes metastable and decays to a stable inactive state. We explore the characteristics of the demand and delivery, as well as the topological properties, which make the distribution network susceptible of failure. An effective temperature is defined, which governs the strength of the activity fluctuations which can induce a collapse. Numerical results, obtained by Monte Carlo simulations of the model on (mainly) scale-free networks, are supplemented with analytic mean-field approximations to the geometrical random field fluctuations and the thermal spin fluctuations. The role of hubs versus poorly connected nodes in initiating the breakdown of network activity is illustrated and related to model parameters.