  We consider weakly interacting jump processes on time-varying random graphs
with dynamically changing multi-color edges. The system consists of a large
number of nodes in which the node dynamics depends on the joint empirical
distribution of all the other nodes and the edges connected to it, while the
edge dynamics depends only on the corresponding nodes it connects. Asymptotic
results, including law of large numbers, propagation of chaos, and central
limit theorems, are established. In contrast to the classic McKean-Vlasov
limit, the limiting system exhibits a path-dependent feature in that the
evolution of a given particle depends on its own conditional distribution given
its past trajectory. We also analyze the asymptotic behavior of the system when
the edge dynamics is accelerated. A law of large number and a propagation of
chaos result is established, and the limiting system is given as independent
McKean-Vlasov processes. Error between the two limiting systems, with and
without acceleration in edge dynamics, is also analyzed.
