  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.
