TITLE: Factorization of a Markov chain as the product of smaller chains
QUESTION [5 upvotes]: Consider a discrete-time Markov chain on $n$ states $S = \{1,2,\ldots,n\}$,  with transition dynamics
$$ \mathbf{x}^{(t+1)} = \mathbf{P} \mathbf{x}^{(t)},$$
where $\mathbf{P}$ is the transition matrix, a stochastic matrix (nonnegative entries and unit column sums). We are interested in the possibility of decomposing this chain into a product of 2 chains on two disjoint subject sets of states $A,B \subseteq S$ (of approx same size),
with dynamics $\mathbf{P}_A$ and $\mathbf{P}_B$ resp. so that
$$\mathbf{x}_A^{(t+1)} = \mathbf{P}_A \mathbf{x}_A^{(t)},\text{ and }\mathbf{x}_B^{(t+1)} = \mathbf{P}_B \mathbf{x}_B^{(t)}, $$
where $\mathbf{P}_A$ and $\mathbf{P}_B$ are stochastic matrices on $\#A$ and $\#B$ states resp. When we can produce such a factorization, we shall say that the original chain is splittable (index $r = 1$), otherwise it is irreducible (index $r = 0$).
Question: I wish to extend / relax this split index $r$ to a general scenario and get a float $r(\mathbf{P}) \in [0, 1]$ which measures how well / bad the original chain can be approximated can be factored into smaller chains. Is there anything in literature (maybe linear algebra ?, graph theory ?) that can shed light on this ?

REPLY [3 votes]: What you need is a definition of graph connectivity for undirected graphs. One can try to generalize the idea of Fiedler vector from directed graphs: if the second-to-last singular value of $I-P$ is zero, then the matrix is "splittable", otherwise it is not. For instance, you can use the ratio $\sigma_{n-1}/\sigma_1$ as a measure of connectivity.
If you have already a candidate approximate splitting into two blocks $\begin{bmatrix}P_{11} & P_{12} \\ P_{21} & P_{22}\end{bmatrix}$, another simple performance measure is $\frac{\|P_{12}\|+\|P_{21}\|}{\|P_{11}\|+\|P_{12}\|+\|P_{21}\|+\|P_{22}\|}$, which also makes sense as "distance from the closest splittable matrix".