Phase Transitions in Transportation Networks with Nonlinearities

TL;DR

This paper studies transportation networks with nonlinearities, revealing phase transitions and complex regimes, including a glassy transition to computational hardness, depending on resource distribution.

Contribution

It introduces a model capturing nonlinear resource competition in transportation networks and analyzes phase transitions and regimes, including a glassy transition, based on resource distribution.

Findings

Different regimes with fractional satisfied nodes resemble Devil's staircase.

Behavior similar to vertex cover and close packing problems observed.

Bimodal resource distribution induces a glassy, hard-to-solve regime.

Abstract

We investigate a model of transportation networks with nonlinear elements which may represent local shortage of resources. Frustrations arise from competition for resources. When the initial resources are uniform, different regimes with discrete fractions of satisfied nodes are observed, resembling the Devil's staircase. We demonstrate how functional recursions are converted to simple recursions of probabilities. Behavior similar to those in the vertex cover or close packing problems are found. When the initial resources are bimodally distributed, increases in the fraction of rich nodes induce a glassy transition, entering an algorithmically hard regime.