Approximating Extremely Large Networks via Continuum Limits

This paper is concerned with modeling of networks with an extremely large number of components using partial differential equations (PDEs). This modeling method is based on the convergence of a sequence of underlying Markov chains of the network indexed by N, the number of components in the network. As N goes to infinity, the sequence converges to a continuum limit, which is the solution of a certain PDE. We provide sufficient conditions for the convergence and characterize the rate of convergence. As an application, we model large wireless sensor networks by PDEs. While traditional Monte Carlo simulation for extremely large networks is practically infeasible, PDEs can be solved with reasonable computation overhead using well-established mathematical tools.