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.


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