  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.
