Counting Vertices in a Voter-type Model

The Neighborhood Attack model is a Voter type model, which takes a finite graph, assigns 1's and -1's to its nodes (vertices), and then runs a Markov chain on the graph by uniformly at random picking a node at every turn, and then switching the values of the node and its neighbors to 1's or -1's according to a (not necessarily fair) coin toss. We show, via a Stein's method argument, that for certain (highly symmetric) families of graphs the number of 1's in the Neighbourhood Attack Voter-type model is asymptotically normally distributed as the number of nodes tends to infinity.