This is actually a more subtle case because it depends on whether or not we identify isomorphic states during the evolution.
If we do identify them, and we keep random walking through the system, we would be reconstructing different subgraphs of the same causal graph. I.e., take a look at what MultiwaySystem
does for this system:
ResourceFunction["MultiwaySystem"][
"WolframModel" -> {{{0, 0}} -> {{0, 1}, {1, 1}}, {{0, 1}, {1,
1}} -> {{0, 0}}}, {{{0, 0}}}, 10, "CausalGraphStructure"]
One can imagine the single-way evolution paths uncovering different parts of that same graph. That of course does not mean we will always get the same graph in the limit of steps going to infinity, but one could get the same graph if one chooses a particular updating sequence after any number of arbitrary events. Note that to make this graph, one had to do an isomorphism between different steps and identify different edges with each other, which WolframModel
does not do.