Flows through random networks

My two current students Shankar Bhamidi and Jason Lenderman and I are working on a range of specific problems within the following general framework. We have some probability model for a network (i.e. a graph with some context-dependent extra structure). We want to route flow simultaneously between each pair of vertices. There may be feasibility constraints (e.g. capacities on edges) and subject to feasibility we seek an optimum flow (w.r.t. to some cost structure). While there is substantial literature on the algorithmic aspects of such flow problems, we take more of a statistical physics approach: we seek to study directly the spatial structure of the optimal flow.

We will eventually write a survey paper. Here are the technical papers so far:

Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities (with C. McDiarmid and A. Scott)
Edge Flows in the Complete Random-Lengths Network (with S. Bhamidi).
Cost-volume relationships for flows through a disordered network
Optimal flow through the disordered lattice
Optimal spatial transportation networks where link-costs are sublinear in link-capacity.

Also look at some recent talks.