Cost-volume relationships for flows through a disordered network

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.