Consider a quantity $x$ diffusing on a graph according to $$\frac{\Bbb dx}{\Bbb dt}=-Lx+a,$$ where $L$ is the graph Laplacian and $a$ encodes sources and sinks. In the steady state, $x_0=L^{-1}a$.
How can I find the current $c$ flowing through each edge in the steady state given the quantity $x_0$?
I see that if every node had degree 1, we would have $c_{ij} \sim x_i-x_j$ but I’m having a hard time finding $c_{ij}$ in the general case.