Balancing traffic in networks: redundancy, learning, and the effect of stochastic fluctuations

Abstract

We study the distribution of traffic in networks whose users try to minimise their delays by adhering to a simple learning scheme inspired by the replicator dynamics of evolutionary game theory. The stable steady states of these dynamics coincide with the network's Wardrop equilibria and form a convex polytope whose dimension is determined by the network's redundancy (an important concept which measures the "linear dependence" of the users' paths). Despite this abundance of stationary points, we show that the long-term behaviour of the replicator dynamics is remarkably simple: every solution orbit converges to a Wardrop equilibrium. On the other hand, a major challenge occurs when the users' delays fluctuate unpredictably due to random external factors. In that case, interior equilibria are no longer stationary, but strict equilibria remain stochastically stable irrespective of the fluctuations' magnitude. In fact, if the network has no redundancy and the users are patient enough, we show that the long-term average of the users' traffic flows converges to the vicinity of an equilibrium, and we also estimate the corresponding invariant distribution.