Let $G$ be a connected, undirected graph, with countably infinite set vertices and countably infinite set of edges. Assume the degree of each vertex is finite.
Let each vertex carry one of two values: $1$ or $-1$.
Now, repeatedly update $G$ by replacing the value at each vertex with the value carried by most of its neighbors (in case of a draw - the value of the vertex remains unchanged). Note that the value at each vertex depends only on the values of its neighbors in the $previous$ state of the whole graph, i.e. the replacement takes place at all vertices simultaneously.
Does there exist such a graph as described, with certain initial values at vertices, that contains a vertex where the value is $not$ eventually constant, after many updates, and not eventually changes with a period of two (e.g. $1$, $-1$, $1$, $-1$...)?