Propagation models on networks: bottom-up

Abstract

In this chapter, we present a different approach to deriving exact models. In Chapter 2, we began with equations for every possible state of the system and then aggregated them into a simpler form. Here, we begin by deriving separate equations for the status of each node. These typically depend on the states of pairs of nodes, so we introduce equations for the pairs, which in turn depend on triples. We build up equations at each level. For a typical network, the number of equations we obtain is too large to be tractable. Consequently, we introduce “closures”, whereby terms corresponding to larger structures are represented in terms of smaller structures, in order to create a closed system of equations. In most cases, this representation involves an approximation, but in the case of SIR dynamics on trees or networks with cut-vertices, it is possible to reduce the number of equations considerably while keeping the model exact.