Markov Process graphical representation

Question

I don't understand how the picture has been constructed. Specifically how $\mu^{11}=-(\mu^{12}+\mu^{13}+\mu^{14})$ and $\mu^{44}=-\mu^{43}$ has been graphically represented. Here $\mu^{ij}$ is the rate at which we change from state $i$ to state $j$.

Answer 1

If $\mu^{ij}$ is nonzero, it means that there is nonzero probability that a transition occurs from state $i$ to state $j$. Each of the states is represented by a vertex in the graph (in the picture, each vertex is drawn as a box). In your example, there are four states ($G$ is a $4\times4$ matrix). Each nonzero $\mu^{ij}$ is represented as a weighted edge in the graph from $i$ to $j$ with weight $\mu^{ij}$.

The elements $\mu^{11}$ and $\mu^{44}$ are not represented in the graph. This is because the diagonal entries are there only to make $G$ a generator matrix and correspond to no transition rate from one state to another. However, some texts display the adjacency graph of $G$, in which diagonal entries are represented as loops (e.g., an edge from $1$ to itself).