  We study the asymptotic behavior of deterministic, continuous-time imitation
dynamics for population games over networks. The basic assumption of this
learning mechanism -- encompassing the replicator dynamics -- is that players
belonging to a single population exchange information through pairwise
interactions, whereby they get aware of the actions played by the other players
and the corresponding rewards. Using this information, they can revise their
current action, imitating the one of the players they interact with. The
pattern of interactions regulating the learning process is determined by a
community structure. First, the set of equilibrium points of such network
imitation dynamics is characterized. Second, for the class of potential games
and for undirected and connected community networks, global asymptotic
convergence is proved. In particular, our results guarantee convergence to a
Nash equilibrium from every fully supported initial population state in the
special case when the Nash equilibria are isolated and fully supported.
Examples and numerical simulations are offered to validate the theoretical
results and counterexamples are discussed for scenarios when the assumptions on
the community structure are not verified.
