Probabilistic Zero Forcing on Random Graphs

TL;DR

This paper investigates probabilistic zero forcing on Erdős-Rényi random graphs, analyzing how the propagation time depends on the edge probability and initial conditions, revealing logarithmic time bounds.

Contribution

It introduces probabilistic zero forcing, extending deterministic zero forcing, and provides probabilistic bounds on propagation time in Erdős-Rényi graphs.

Findings

Propagation time is about (1+o(1)) log_2log_2 n ounds when p=log^{-o(1)} n.

Propagation time is  heta(log(1/p)) rounds when log n/n \u2264 p log^{-O(1)} n.

High probability results depend on the edge probability p and initial blue vertex placement.

Abstract

Zero forcing is a deterministic iterative graph coloring process in which vertices are colored either blue or white, and in every round, any blue vertices that have a single white neighbor force these white vertices to become blue. Here we study probabilistic zero forcing, where blue vertices have a non-zero probability of forcing each white neighbor to become blue. We explore the propagation time for probabilistic zero forcing on the Erd\H{o}s-R\'eyni random graph $\Gnp$ when we start with a single vertex colored blue. We show that when $p=\log^{-o(1)}n$, then with high probability it takes $(1+o(1))\log_2\log_2n$ rounds for all the vertices in $\Gnp$ to become blue, and when $\log n/n\ll p\leq \log^{-O(1)}n$, then with high probability it takes $\Theta(\log(1/p))$ rounds.