Electrical network

An electrical network is a weighted graph whose edge weights are interpreted as electrical conductances. Random-walk hitting quantities can then be expressed through voltages, currents, and effective resistance.

The effective resistance between vertices $a$ and $z$ is the voltage difference required to send one unit of current from $a$ to $z$.

Thomson principle says that effective resistance is the minimum energy among unit flows from $a$ to $z$:

If ${\Pi }_1,\dots ,{\Pi }_m$ are pairwise edge-disjoint cutsets separating $a$ from $z$ in an electrical network, then

For a random walk on a finite electrical network,