This is nice. So, follow-up question. In economics, there's "input-output models" which amount to the summation of adjacency matrices for inter-sectoral demand

This is nice. So, follow-up question. In economics, there's "input-output models" which amount to the summation of adjacency matrices for inter-sectoral demand linkages such that S = I + A + A^2 + A^3+... = (I-A)^(-1) We know this to be true because matrix A is markovian, so it's a bounded operator, so we have a Neumann series. What happens if we have probability distributions for the elements of A? This is the context where it arises: we define a vector of x interindustrial demands that must satisfy itself through production coefficients A (the weights in the graph of cross-sector dependencies), such that x = Ax but then we give this balanced system an additive shock ("consumer demands") b so that x' = x+e = Ax+b but we still want to satisfy Ax' = x'. So: A(x+e) = Ax+b => A*e = b => e = (I-A)^(-1) b This is the shock to the interindustrial demands provoked by b. We have to know that (I-A) is invertible, which by the Perron-Frobenius theorem is true if A happens to be markovian. This matters because the interindustrial linkage graph A is very noisy (it's hammered into place at the national accounts offices from disperse information), and much basic demand-shock appraisal (in a multi-sector context at least) still uses this basic ("Leontieff") model; and more sophisticated models apply variants of this too; but it's very hard to get anything said about confidence intervals -- other than by means of simulation. So -- maybe it's an interesting, applied problem to work on!