Monotone convergence of spreading processes on networks

We analyze the Bass and SI models for the spreading of innovations and epidemics, respectively, on homogeneous complete networks, on one-dimensional networks, and on heterogeneous two-groups complete networks. We allow the network parameters to be time dependent, which is a prerequisite for the analysis of optimal promotional strategies on networks. Using a novel top-down analysis of the master equations, we present a simple proof for the monotone convergence of these models to their respective infinite-population limits. This leads to explicit expressions for the expected adoption or infection level in the Bass and SI models with time-dependent parameters on infinite homogeneous complete and circular networks, and on heterogeneous two-groups complete networks.