Cost-volume relationships for flows through a disordered network

TL;DR

This paper develops a probabilistic approach using the cavity method to analyze the relationship between flow volume and minimum cost in large, disordered networks with nonlinear edge costs.

Contribution

It introduces a novel, probabilistic framework based on the cavity method to compute cost-volume relationships in large disordered networks, extending previous models.

Findings

Method to compute cost-volume function in infinite-size networks

Uses a probabilistic reformulation of the cavity method

Applicable to various flow problems on random networks

Abstract

In a network where the cost of flow across an edge is nonlinear in the volume of flow, and where sources and destinations are uniform, one can consider the relationship between total volume $v$ of flow through the network and the minimum cost $c = Psi(v)$ of any flow with volume $v$. Under a simple probability model (locally tree-like directed network, independent cost-volume functions or different edges) we show how to compute $\Psi(v)$ in the infinite-size limit. The argument uses a probabilistic reformulation of the cavity method from statistical physics, and is not rigorous as presented here. The methodology seems potentially useful for many problems concerning flows on this class of random networks.