Dynamic monopolies with randomized starting configuration

Properties of systems with majority voting rules have been exhaustingly studied. In this work we focus on the randomized case where the system is initialized by randomized initial set of seeds. Our main aim is to give an asymptotic estimate for sampling probability, such that the initial set of seeds is (is not) a dynamic monopoly almost surely. After presenting some trivial examples, we present exhaustive results for toroidal mesh and random 4-regular graph under simple majority scenario.