  We consider a dynamical process in a network which distributes all particles
(tokens) located at a node among its neighbors, in a round-robin manner.
  We show that in the recurrent state of this dynamics (i.e., disregarding a
polynomially long initialization phase of the system), the number of particles
located on a given edge, averaged over an interval of time, is tightly
concentrated around the average particle density in the system. Formally, for a
system of $k$ particles in a graph of $m$ edges, during any interval of length
$T$, this time-averaged value is $k/m \pm \widetilde{O}(1/T)$, whenever
$\gcd(m,k) = \widetilde{O}(1)$ (and so, e.g., whenever $m$ is a prime number).
To achieve these bounds, we link the behavior of the studied dynamics to
ergodic properties of traversals based on Eulerian circuits on a symmetric
directed graph. These results are proved through sum set methods and are likely
to be of independent interest.
  As a corollary, we also obtain bounds on the \emph{idleness} of the studied
dynamics, i.e., on the longest possible time between two consecutive
appearances of a token on an edge, taken over all edges. Designing trajectories
for $k$ tokens in a way which minimizes idleness is fundamental to the study of
the patrolling problem in networks. Our results immediately imply a bound of
$\widetilde{O}(m/k)$ on the idleness of the studied process, showing that it is
a distributed $\widetilde{O}(1)$-competitive solution to the patrolling task,
for all of the covered cases. Our work also provides some further insights that
may be interesting in load-balancing applications.
