  In this work we establish that finite directed graphs give rise to semiflows
on the power set of their nodes. We analyze the topological dynamics for
semiflows on finite directed graphs by characterizing Morse decompositions,
recurrence behavior and attractor-repeller pairs under weaker assumptions. As
is expected, the discrete metric plays an important role in our constructions
and their consequences. The connections between the semiflow, graph theory and
Markov chains are here explored. We lay the foundation for a dynamical systems
approach to hybrid systems with Markov chain type perturbations.
