Propagation time for weighted zero forcing

Abstract: Zero forcing is a graph coloring process that was defined as a tool for bounding the minimum rank and maximum nullity of a graph. It has also been used for studying control of quantum systems and monitoring electrical power networks. One of the problems from the 2017 AIM workshop "Zero forcing and its applications" was to explore edge-weighted probabilistic zero forcing, where edges have weights that determine the probability of a successful force if forcing is possible under the standard zero forcing coloring rule. In this paper, we investigate the expected time to complete the weighted zero forcing coloring process, known as the expected propagation time, as well as the time for the process to be completed with probability at least $\alpha$, known as the $\alpha$-confidence propagation time. We demonstrate how to find the expected and confidence propagation times of any edge-weighted graph using Markov matrices. We also determine the expected and confidence propagation times for various families of edge-weighted graphs including complete graphs, stars, paths, and cycles.