I considered the problem of a form of CTMC evolving in a graph:

Consider a graph of $G(V,E)$ with $|V|=N$ nodes. Each node has a 1-0 CTMC associated with it:

- There is a vertex dependent rate $\mu_i$ such that node $i$ moves from 1 to 0 at a Poisson rate $\mu_i$.
- There is an edge-dependent Poisson rate $\lambda_{ij}$ along the edge $e(i,j)\in E$ which takes $j$ from 0 to 1 whenever node $i$ is in state 1.
- As long as node $j$ is in state 1, its neighbours in state 0 could each move to state 1 at a Poisson rate of $\lambda_{jk}$ where $k$ is one of the neighbours of $j$. A node in state 0 cannot influence the states of its neighbouring nodes.
- If node $j$ is in state 0, and nodes $i$ and $k$ are its neighbours in state 1, the probabilities of $i$ changing $j$'s state and of $k$ changing $j$'s state are mutually exclusive.

In such a setting, if $X(0)$ represents the set of nodes in state 1 at time 0 and $T_j$ the random variable of the first time when $j$ moves into state 1, how do we determine $E(T_j|X(0)=\{i\})$?

Could someone provide me with relevant literature pertaining to such problems?