Sensitivity Analysis of Network Traffic Equilibrium Revisited: The Corrected Approach

Abstract

This paper resolves some recent puzzles and criticisms of the gradient-based sensitivity analysis of network traffic equilibria originally proposed by Tobin and Friesz, and develops a simple but general approach for calculating the derivatives of equilibrium link flows. The paper shows that, under the conditions that the link cost function vector is strongly monotone and there exists a set of strictly positive flows on the minimum cost or equilibrated paths, the classical implicit function theorem can always be applied for any set of maximum number of linearly independent and established paths. As a result, the equilibrium link flows are differentiable under these mild conditions and their derivatives can always be obtained by the formula developed from the implicit function theorem, without dependence on the network topology. The paper shows that non-differentiability of the equilibrium link flows may occur only at a degenerate equilibrium point where there exists at least on degenerate, equilibrated path on which flow must be zero. The paper will further show that the gradient formula, when applied to a degenerate equilibrium point, can provide the left and right derivatives of link flows (if not differentiable) and the desired derivatives (if differentiable) with respect to a perturbation parameter when appropriately including or not including the degenerate path(s). Analytical examples facilitate understanding of the simplicity and generality of the corrected gradient formula and the flaws inherent in the old sensitivity analysis approach.

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