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We consider congestion games on networks with nonatomic users anduser-specific costs. We are interested in the uniqueness property defined byMilchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness ofequilibrium in networks. Math. Oper. Res. 30 225-244] as the uniqueness ofequilibrium flows for all assignments of strictly increasing cost functions. Hesettled the case with two-terminal networks. As a corollary of his result, itis possible to prove that some other networks have the uniqueness property aswell by adding common fictitious origin and destination.  In the present work, we find a necessary condition for networks with severalorigin-destination pairs to have the uniqueness property in terms of excludedminors or subgraphs. As a key result, we characterize completely bidirectionalrings for which the uniqueness property holds: it holds precisely for ninenetworks and those obtained from them by elementary operations. For otherbidirectional rings, we exhibit affine cost functions yielding to two distinctequilibrium flows. Related results are also proven. For instance, wecharacterize networks having the uniqueness property for any choice oforigin-destination pairs.

The uniqueness property for networks with several origin-destination pairs