Explicit solutions of the SI and Bass models on sparse Erd\H{o}s-R\'enyi and regular networks

TL;DR

This paper derives explicit, analyzable solutions for the expected adoption and infection levels in SI and Bass models on large sparse Erdős-Rényi and regular networks, demonstrating their accuracy as network size grows.

Contribution

It provides the first explicit solutions for these models on such networks, showing their validity in the large network limit.

Findings

Explicit solutions are derived for large sparse networks.

Cycle effects vanish as network size increases.

Solutions are exact in the limit of infinite network size.

Abstract

We derive explicit expressions for the expected adoption and infection level in the Bass and SI models, respectively, on sparse Erd\H{o}s-R\'enyi networks and on $d$-regular networks. These expressions are soloutions of first-order ordinary differential equations, which are fairly easy to analyze. To prove that these expressions are exact, we show that the effect of cycles vanishes as the network size goes to infinity.