Let's say I have $M=\begin{bmatrix}1&2&1\\ 4&2&0 \\ 1& 1& 1\end{bmatrix}$, a $3\times3$ matrix which is the transition matrix or adjacency matrix of a $3$-node graph.
I would like to merge node $1$ and $2$ into a new $12$ node, and thus only have a $2$-node graph but (there will be loss of info of course) that graph will represent the same "population" dynamics.
Gut feeling would be to weight transitions OUT OF $1$ and $2$ by their relative abundance ($38.1$% and $61.9$% in that case) and to add the transitions TO $1$ and $2$. In that case I would get $\begin{bmatrix}4,67&1 \\ 1&1\end{bmatrix}$.
And indeed I can see that then the growth rate (max eigenvalue) is the same for both graphs. That the abundance (right eigenvector) or stable-age-population are the same too.
However left (or left*right) eigenvector are different, and more importantly from an interpretation standpoint: the sensitivity of growth rate to say the transition of node $3$ to node $3$. (The only unchanged number in my graph/matrix) is different ($0.075$ vs $0.069$.)
Those graphs are not really "equivalent" or "similar" then?
Any idea of how to improve that? Another way of merging those $2$ nodes, whilst (not artificially) keeping the sensitivity to $M_{3,3}$?
Thanks a lot.