The uniqueness property for networks with several origin–destination pairs

Abstract

We consider congestion games on networks with nonatomic users and user-specific costs. We are interested in the uniqueness property defined by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions. He settled the case with two-terminal networks. As a corollary of his result, it is possible to prove that some other networks have the uniqueness property as well by adding common fictitious origin and destination. In the present work, we find a necessary condition for networks with several origin-destination pairs to have the uniqueness property in terms of excluded minors or subgraphs. As a key result, we characterize completely bidirectional rings for which the uniqueness property holds: it holds precisely for nine networks and those obtained from them by elementary operations. For other bidirectional rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. Related results are also proven. For instance, we characterize networks having the uniqueness property for any choice of origin-destination pairs. and those obtained from them by elementary operations. For other rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. It allows to describe infinite families of graphs for which the uniqueness property does not hold. For instance, there is a family of ring networks such that every network with a minor in this family does not have the uniqueness property. Preliminaries on graphsAn undirected graph is a pair G = (V, E) where V is a finite set of vertices and E is a family of unordered pairs of vertices called edges. A directed graph, or digraph for short, is a pair D = (V, A) where V is a finite set of vertices and A is a family of ordered pairs of vertices called arcs. A mixed graph is a graph having edges and arcs. More formally, it is a triple M = (V, E, A) where V is a finite set of vertices, E is a family of unordered pairs of vertices (edges) and A is a family of ordered pairs of vertices (arcs). Given an undirected graph G = (V, E), we define the directed version of G as the digraph D = (V, A) obtained by replacing each (undirected) edge in E by two (directed) arcs, one in each direction. An arc of G is understood as an arc of its directed version. In these graphs, loops -edges or arcs having identical endpoints -are not allowed, but pairs of vertices occuring more than onceparallel edges or parallel arcs -are allowed.A walk in a directed graph D is a sequenceIf all v i are distinct, the walk is called a path. If no confusion may arise, we identify sometimes a path P with the set of its vertices or with the set of its arcs, allowing to use the notation v ∈ P (resp. a ∈ P ) if a vertex v (resp. an arc a) occurs in P . An undirected graph G = (V , E ) is a subgraph of an undirected graph G = (V, E) if V ⊆ V and E ⊆ E. An undirected graph G is a minor of an undirected graph G if G is obtained by contracting edges (poss...

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