Parameter-space analysis of functional path-flow substitutions in Wardrop network equilibrium models

Abstract

We study a path-flow Wardrop network equilibrium model in which total demand is split among paths subject to non-negativity, demand conservation, and a cost-balance condition. Each path flow is reparameterized as q_p = ϕ(s_p) for a scalar  function ϕ and a parameter s_p, and we build a single mathematical framework for comparing six choices of ϕ: cosine, sine, the exponential, the hyperbolic sine, the hyperbolic cosine, and the shifted hyperbolic cosine ϕ(s) = cosh(s) - 1. For each substitution we prove a feasibility bound, identify the admissible domain on which it is injective, give its closed-form inverse, and derive the dimensionless condition number k_p = |ϕ'(s_p) s_p/q_p|. We prove --- conditionally, not universally --- that if two admissible substitutions produce the same flow vector q^*, then the link flow, costs, and all equilibrium quantities are identical; the substitutions differ in feasibility, in the zero-flow property (ability to represent an unused path at a finite parameter), in parameter uniqueness, and in k_p. A fully worked example on a four-node, five-link communication network with a naturally unused path illustrates the theory with reproducible, independently verified computations.