Decay of correlation in network flow problems

Abstract

We develop a general theory for the local sensitivity of optimal points of constrained network optimization problems under perturbations of the constraints. For the network flow problem, we show that local perturbations on the constraints have an impact on the components of the optimal point that decreases exponentially with the graph-theoretical distance. The exponential rate is controlled by the spectral radius of a sub-stochastic transition matrix of a killed random walk associated to the network. For graphs where this spectral radius is well-behaved (bounded, for instance) as a function of the dimension of the network, our theory yields the first-known incarnation of the decay of correlation principle in constrained optimization.