Probabilistic Zero Forcing on Grid, Regular, and Hypercube Graphs

TL;DR

This paper analyzes probabilistic zero-forcing on various graph families, deriving asymptotic bounds for the expected propagation time, including optimal bounds for grid graphs, bounds for regular graphs, and hypercube graphs.

Contribution

It establishes new asymptotic bounds for the expected propagation time of probabilistic zero-forcing on grid, regular, and hypercube graphs, including optimal and upper bounds.

Findings

Optimal bound of Θ(m+n) for grid graphs

Upper bound of O((log d)/d * n) for d-regular graphs

Upper bound of O(n log n) for hypercube graphs

Abstract

Probabilistic zero-forcing is a coloring process on a graph. In this process, an initial set of vertices is colored blue, and the remaining vertices are colored white. At each time step, blue vertices have a non-zero probability of forcing white neighbors to blue. The expected propagation time is the expected amount of time needed for every vertex to be colored blue. We derive asymptotic bounds for the expected propagation time of several families of graphs. We prove the optimal asymptotic bound of $\Theta(m+n)$ for $m\times n$ grid graphs. We prove an upper bound of $O \left(\frac{\log d}{d} \cdot n \right)$ for $d$-regular graphs on $n$ vertices and provide a graph construction that exhibits a lower bound of $\Omega \left(\frac{\log \log d}{d} \cdot n \right)$. Finally, we prove an asymptotic upper bound of $O(n \log n)$ for hypercube graphs on $2^n$ vertices.