Mean-field Approximation for Stochastic Population Processes in Networks under Imperfect Information

This paper studies a general class of stochastic population processes in which agents interact with one another over a network. Agents update their behaviors in a random and decentralized manner based only on their current state and the states of their neighbors. It is well known that when the number of agents is large and the network is a complete graph (has all-to-all information access), the macroscopic behavior of the population converges to a differential equation called a {\it mean-field approximation}. When the network is not complete, it is unclear in general whether there exists a suitable mean-field approximation for the macroscopic behavior of the population. This paper provides general conditions on the network and policy dynamics for which a suitable mean-field approximation exists. First, we show that as long as the network is well-connected, the macroscopic behavior of the population concentrates around the {\it same} mean-field system as the complete-graph case. Next, we show that as long as the network is sufficiently dense, the macroscopic behavior of the population concentrates around a mean-field system that is, in general, {\it different} from the mean-field system obtained in the complete-graph case. Finally, we provide conditions under which the mean-field approximation is equivalent to the one obtained in the complete-graph case.