Gradually Declining Immunity Retains the Exponential Duration of Immunity-Free Diffusion

TL;DR

This paper introduces the cSIRS process with gradually declining immunity and demonstrates that its survival time on certain networks closely resembles the no-immunity SIS process, challenging assumptions about immunity effects.

Contribution

The paper presents the cSIRS process, modeling gradually declining immunity, and rigorously analyzes its survival time, showing it behaves similarly to the no-immunity SIS process on key network types.

Findings

Expected survival time of cSIRS is similar to SIS on star graphs.

Expected survival time of cSIRS is similar to SIS on expanders.

Gradually declining immunity effectively acts like no immunity.

Abstract

Diffusion processes pervade numerous areas of AI, abstractly modeling the dynamics of exchanging, oftentimes volatile, information in networks. A central question is how long the information remains in the network, known as survival time. For the commonly studied SIS process, the expected survival time is at least super-polynomial in the network size already on star graphs, for a wide range of parameters. In contrast, the expected survival time of the SIRS process, which introduces temporary immunity, is always at most polynomial on stars and only known to be super-polynomial for far denser networks, such as expanders. However, this result relies on featuring full temporary immunity, which is not always present in actual processes. We introduce the cSIRS process, which incorporates gradually declining immunity such that the expected immunity at each point in time is identical to that of the SIRS process. We study the survival time of the cSIRS process rigorously on star graphs and expanders and show that its expected survival time is very similar to that of the SIS process, which features no immunity. This suggests that featuring gradually declining immunity is almost as having none at all.