Theorem 9.10 -- Thomson's principle

Let $c$ be a network on a finite vertex set: a symmetric nonnegative conductance function with total conductance $c(x)=\sum_yc(x,y)>0$ at every vertex, whose associated walk $P(x,y)=c(x,y)/c(x)$ is irreducible. Fix distinct vertices $a\ne z$. A flow from $a$ to $z$ is an antisymmetric edge function $\theta(x,y)=-\theta(y,x)$ vanishing wherever $c$ does, satisfying the node law $\sum_y\theta(x,y)=0$ at every vertex except $a$ and $z$; its strength is the net flux $\sum_y\theta(a,y)$ out of $a$, and a flow of strength one is a unit flow. The energy of a flow is

each undirected edge counted once. The effective resistance $R(a\leftrightarrow z)$ is defined through the voltage $W(x)=\mathbb P_x\{\tau_a<\tau_z\}$ and the current $\|I\|=\sum_yc(a,y)[W(a)-W(y)]$ as $R(a\leftrightarrow z)=\|I\|^{-1}$.

The theorem (Thomson's Principle, Theorem 9.10 of Levin–Peres–Wilmer) asserts:

1. $R(a\leftrightarrow z)=\inf\{\mathcal E(\theta):\theta\ \text{a unit flow from}\ a\ \text{to}\ z\}$;
2. the infimum is attained — some unit flow (the current flow) has energy exactly $R(a\leftrightarrow z)$.

Resistance is a variational quantity: any unit flow certifies an upper bound on it, which is the source of all flow-based hitting-time estimates.