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We study the distribution of traffic in networks whose users try to minimisetheir delays by adhering to a simple learning scheme inspired by the replicatordynamics of evolutionary game theory. The stable steady states of thesedynamics coincide with the network's Wardrop equilibria and form a convexpolytope whose dimension is determined by the network's redundancy (animportant concept which measures the "linear dependence" of the users' paths).Despite this abundance of stationary points, the long-term behaviour of thereplicator dynamics turns out to be remarkably simple: every solution orbitconverges to a Wardrop equilibrium.  On the other hand, a major challenge occurs when the users' delays fluctuateunpredictably due to random external factors. In that case, interior equilibriaare no longer stationary, but strict equilibria remain stochastically stableirrespective of the fluctuations' magnitude. In fact, if the network has noredundancy and the users are patient enough, we show that the long-termaverages of the users' traffic flows converge to the vicinity of anequilibrium, and we also estimate the corresponding invariant measure.

Balancing Traffic in Networks: Redundancy, Learning and the Effect of Stochastic Fluctuations