  Consider a network of $n$ single-server queues where tasks arrive
independently at each of the servers at rate $\lambda_n$. The servers are
interconnected by a graph that is resampled at rate $\mu_n$ in a way that is
symmetric with respect to the servers, and each task is dispatched to the
shortest queue in the graph neighborhood where it appears. The so-called
occupancy process describes the empirical distribution of the number of tasks
across the servers. This stochastic process evolves on the underlying dynamic
graph, and its dynamics depend on the the number of tasks at each individual
server and the neighborhood structure of the graph. We prove that this
dependency disappears in the limit as $n \to \infty$ when $\lambda_n / n \to
\lambda$ and $\mu_n \to \infty$, and establish that the limit of the occupancy
process is given by a system of differential equations that depends solely on
$\lambda$ and the limiting degree distribution of the graph. We further show
that the stationary distribution of the occupancy process converges to an
equilibrium point of the differential equations, and derive properties of this
equilibrium that reflect the impact of the degree distribution. Our focus is on
truly sparse graphs where the maximum degree is uniformly bounded across $n$,
making neighboring servers strongly correlated.
