 
Summary: MATHEMATICS OF OPERATIONS RESEARCH
Vol. 33, No. 4, November 2008, pp. 769786
issn 0364765X eissn 15265471 08 3304 0769
informs®
doi 10.1287/moor.1080.0327
© 2008 INFORMS
CostVolume Relationship for Flows Through
a Disordered Network
David J. Aldous
Department of Statistics, University of California, Berkeley, California 94720,
aldous@stat.berkeley.edu
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 of flow through the network and the minimum cost of any flow
with given volume. Under a simple probability model (locally treelike directed network, independent costvolume functions
for different edges) we show how to compute the minimum cost in the infinitesize 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. Making arguments rigorous is a
challenging open problem.
Key words: capacitated network; cavity method; first passage percolation; network flow; probability model
MSC2000 subject classification: Primary: 90B15; secondary: 60K30
