Random processes on coloured graphs

Question

Let a random process on a $k$-coloured graph be any set of rules that determine how nodes change their colour probabilistically from time step to time step, depending only on the colours of their neighbours.

This very much looks like stochastic cellular automata but differs from them in that the underlying graph is not a regular infinite grid, but an arbitrary finite graph. This graph is not supposed to be genuinely random but may be highly regular, even vertex-transitive. At least the graphs under consideration are supposed to be quite large, with at least say 100 or 1,000 or 10,000 nodes.

A currently rather interesting example are $4$-coloured graphs with colours $\color{green}S$ (for susceptible), $\color{gold}E$ (for exposed), $\color{red}I$ (for infectious), and $\color{blue}R$ (for recovered). The corresponding SEIR transition rules are:

1. $\color{green}S\rightarrow \color{green}S$: When a node $v_i$ is $\color{green}S$ and none of its neighbours is $\color{red}I$ at time $t$, then $v_i$ will be $\color{green}S$ at time $t+1$.

2. $\color{green}S\rightarrow \color{gold}E$: When a node $v_i$ is $\color{green}S$ and one of its neighbours is $\color{red}I$ at time $t$, then $v_i$ will be $\color{gold}E$ at time $t+1$ with fixed probability $\tau$ (the transmissibility).

3. $\color{gold}E\rightarrow \color{red}I$: When a node $v_i$ is $\color{gold}E$ at time $t$, then $v_i$ will be $\color{red}I$ at time $t+1$.

4. $\color{red}I\rightarrow \color{red}I$: When a node $v_i$ is $\color{gold}E$ at time $t-1$ and $\color{red}I$ at time $t$, then $v_i$ will be $\color{red}I$ at time $t+1$.

5. $\color{red}I\rightarrow \color{blue}R$: When a node $v_i$ is $\color{red}I$ at times $t-1$ and $t$, then $v_i$ will be $\color{blue}R$ at time $t+1$.

6. $\color{blue}R\rightarrow \color{blue}R$: When a node $v_i$ is $\color{blue}R$ at time $t$, then $v_i$ will be $\color{blue}R$ at time $t+1$.

Given any set $\mathsf{R}$ of probabilistic $k$-colour rules and any initial $k$-coloured graph $G$ one may ask the following questions:

1. Does the process always yield a fixed state with all node colours $c_i(t+1) = c_i(t)$?
   _{For the SEIR rules it obviously does.}

2. When on the average does the process yield the first maximum of the number of $X$-coloured nodes and how large is it on the average?
   _{For the SEIR rules let $X = \color{red}I$.}

3. Assuming that the process yields a fixed final state: What on the average is the number of $X$-coloured nodes in the final state?
   _{For the SEIR rules let $X = \color{blue}R$.}

The considerations become more tractable when you don't choose arbitrarily coloured initial graphs but for example only those with a fixed small number of $X$-coloured nodes, all the others having colour $Y$.
_{For the SEIR rules let $X = \color{gold}E$ and $Y = \color{green}S$.}

All these questions can be quantitatively answered by running the process $N\rightarrow \infty$ times on one and the same graph $G$. The resulting numbers are $\mathsf{R}$-dependent characteristics of $G$.

An interesting question then might be, how these characteristics depend not only on the rules $\mathsf{R}$, but also on classical graph characteristics of the uncoloured graph $G$ like the distributions of degree, eccentricity, and cluster coefficient, and especially regularities and symmetries. (See above: I don't want to consider random graphs only.)

Is there a general and comprehensive theory, introduction and textbook devoted to this specific topic?

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This question originated in another one I asked some time ago: How does the reproduction number depend on characteristics of the physical contact graph of a population? I tried to formulate it more mathematically and more generally.