  Zero forcing is a coloring game played on a graph that was introduced more
than ten years ago in several different applications. The goal is to color all
the vertices blue by repeated use of a (deterministic) color change rule.
Probabilistic zero forcing was introduced by Kang and Yi in [Probabilistic zero
forcing in graphs, Bull. Inst. Combin. Appl. 67 (2013), 9--16] and yields a
discrete dynamical system, which is a better model for some applications. Since
in a connected graph any one vertex can eventually color the entire graph blue
using probabilistic zero forcing, the expected time to do this is a natural
parameter to study. We determine expected propagation time exactly for paths
and cycles, establish the asymptotic value for stars, and present asymptotic
upper and lower bounds for any graph in terms of its radius and order. We apply
these results to obtain values and bounds on $\ell$-round probabilistic zero
forcing, throttling number for probabilistic zero forcing, and confidence
levels for propagation time.
