The Equilibrium Worth of a Network Link

Abstract

Select an arbitrary link T on a directed network whose link costs are fixed, unknown, differentiable disutility functions of link flows (with a symmetric Jacobian). We induce graded variations in the “intrinsic” cost of T due to its own flow by varying the coefficient vector α of a control polynomial pT of this flow. Our sensitivity analysis of such perturbations on the unevaluated equilibrium cost C, for any O(rigin)/D(estination)-pair with arbitrary demand, focuses on the marginal contribution to C of T's equilibrium flow UT. This “shadow price” of UT, as well as the gradient components of C relative to α, are all positive multiples—identical for every O/D-pair-of the rate of change of UT relative to that O/D-pair's demand; and they can all be determined from output-data obtained locally at T. For affine link costs, the shadow price of UT remains constant on every utilized (sub)network as α and O/D-demand both vary, and T-local data then yields a polygonal map with which to predict the resulting link utilization patterns and the variations of C at large.