Monotone convergence of spreading processes on networks

TL;DR

This paper proves that the Bass and SI spreading models on various networks converge monotonically to their limits, providing explicit formulas for expected adoption and infection levels, even with time-varying parameters.

Contribution

It introduces a novel top-down analysis of master equations to establish monotone convergence and derive explicit expressions for spreading processes on diverse networks.

Findings

Models converge monotonically to their limits.

Explicit formulas for expected levels on different networks.

Applicable to time-dependent network parameters.

Abstract

We analyze the Bass and SI models for the spreading of innovations and epidemics, respectively, on homogeneous complete networks, circular networks, and heterogeneous complete networks with two homogeneous groups. We allow the network parameters to be time dependent, which is a prerequisite for the analysis of optimal 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, respectively, on infinite homogeneous complete and circular networks, and on heterogeneous complete networks with two homogeneous groups with time-dependent parameters.