I am modeling some type of social interaction, and came up with the following natural question. Let $K_n$ be the complete graph on $n$ vertices, with some initial edge labeling in some alphabet $A$.

Let $R : A^2 \to A^2$ be a set of relabeling rules, that is, the input is an edge with labels $(a_1,a_2)$ and $R$ relabels this to $(b_1,b_2)$. If the graph is undirected, we assume that $R$ is symmetric in the appropriate manner.

Then, we choose a random (uniformly) total ordering of all edges of $K_n$, and for each edge apply the rule determined by $R$ in this order. We get a new labeling of $K_n$.

It is then natural to let $X_a$ be the random variable counting the number of vertices in the final labeling with label $a$, and ask about $E[X_a]$ or if $X_n$ after scaling is a normal distribution.

**Has anyone studied this model before?**

In this particular example, one can think of edges as interactions between people, and they change state depending on the information they exchange.