This is nice. So, follow-up question.<p>In economics, there's "input-output models" which amount to the summation of adjacency matrices for inter-sectoral demand linkages such that<p>S = I + A + A^2 + A^3+... = (I-A)^(-1)<p>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?<p>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<p>x = A<i>x<p>but then we give this balanced system an additive shock ("consumer demands") b so that<p>x' = x+e = A</i>x+b<p>but we still want to satisfy Ax' = x'. So:<p>A<i>(x+e) = A</i>x+b => A*e = b => e = (I-A)^(-1) b<p>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.<p>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.<p>So -- maybe it's an interesting, applied problem to work on!