Probabilistic zero forcing on random graphs

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 Erdős–Réyni random graph G(n,p) 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))log2log2n rounds for all the vertices in G(n,p) to become blue, and when logn∕n≪p≤log−O(1)n, then with high probability it takes Θ(log(1∕p)) rounds.