Selfish routing revisited: degeneracy, evolution and stochastic fluctuations
Résumé
We study the traffic routing problem in networks whose users try to minimize their latencies by employing a distributed learning rule inspired by the replicator dynamics of evolutionary game theory. The stable states of these dynamics coincide with the network's (Wardrop) equilibrium points and we find that they form a convex polytope whose dimension is determined by the network's degeneracy index (an important concept which measures the overlap of the users' paths). Still, despite this abundance of stable states, we find that (almost) every solution trajectory converges to an equilibrium point at an exponential rate. On the other hand, a major challenge occurs when network latencies fluctuate unpredictably due to random exogenous factors. In that case, we show that the time-average of the traffic flows of sufficiently patient users is still concentrated in a neighborhood of evolutionarily stable equilibria and we estimate analytically the corresponding stationary distribution and convergence times.