I have the following problem, which is motivated by geometric diffusion on a directed graph.
Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} < 1$. Define $$B = (I-A)^{-1} = \sum^\infty_{k=0} A^k.$$ Then $$\frac{B_{ii}}{B_{ji}} \geq \frac{B_{ik}}{B_{jk}}, \quad \forall i,j,k.$$
It seems to hold numerically, but I haven't been able to prove it. Is this a known result?