Network Decontamination with a Single Agent

TL;DR

This paper introduces the concept of immunity number for network graphs to model virus spread and decontamination, demonstrating that nonmonotonic strategies outperform traditional monotonic approaches in certain cases.

Contribution

It defines the immunity number for graphs, analyzes it for various network topologies, and shows that nonmonotonic strategies can achieve better bounds than monotonic ones.

Findings

Upper bounds on immunity number for specific graph classes

Matching lower bounds in some cases

Nonmonotonic strategies outperform monotonic strategies

Abstract

Faults and viruses often spread in networked environments by propagating from site to neighboring site. We model this process of {\em network contamination} by graphs. Consider a graph $G=(V,E)$, whose vertex set is contaminated and our goal is to decontaminate the set $V(G)$ using mobile decontamination agents that traverse along the edge set of $G$. Temporal immunity $\tau(G) \ge 0$ is defined as the time that a decontaminated vertex of $G$ can remain continuously exposed to some contaminated neighbor without getting infected itself. The \emph{immunity number} of $G$, $\iota_k(G)$, is the least $\tau$ that is required to decontaminate $G$ using $k$ agents. We study immunity number for some classes of graphs corresponding to network topologies and present upper bounds on $\iota_1(G)$, in some cases with matching lower bounds. Variations of this problem have been extensively studied in literature, but proposed algorithms have been restricted to {\em monotone} strategies, where a vertex, once decontaminated, may not be recontaminated. We exploit nonmonotonicity to give bounds which are strictly better than those derived using monotone strategies.