Continuum modeling and control of large nonuniform networks

Abstract

Recent research has shown that some Markov chains modeling networks converge to continuum limits, which are solutions of partial differential equations (PDEs), as the number of the network nodes approaches infinity. Hence we can approximate such large networks by PDEs. However, the previous results were limited to uniform immobile networks with a fixed transmission rule. In this paper we first extend the analysis to uniform networks with more general transmission rules. Then through location transformations we derive the continuum limits of nonuniform and possibly mobile networks. Finally, by comparing the continuum limits of corresponding nonuniform and uniform networks, we develop a method to control the transmissions in nonuniform and mobile networks so that the continuum limit is invariant under node locations, and hence mobility. This enables nonuniform and mobile networks to maintain stable global characteristics in the presence of varying node locations.