Using Markov chains to determine expected propagation time for probabilistic zero forcing

TL;DR

This paper models probabilistic zero forcing on graphs using Markov chains to derive an exact formula for expected propagation time and provides bounds for different graph families.

Contribution

It introduces a Markov chain framework to compute exact expected propagation times in probabilistic zero forcing on graphs.

Findings

Derived an exact formula for expected propagation time using Markov chains.

Established bounds on propagation time for various graph families.

Demonstrated the applicability of Markov models to analyze probabilistic zero forcing.

Abstract

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact formula for expected propagation time. We apply Markov chains to determine bounds on expected propagation time for various families of graphs.