Theorem 9.12 -- Rayleigh's monotonicity law

Let $c$ and $c'$ be two networks on the same finite vertex set — symmetric nonnegative conductance functions, each with strictly positive total conductance $c(x)=\sum_yc(x,y)$ at every vertex and an irreducible associated walk $P(x,y)=c(x,y)/c(x)$. For distinct vertices $a\ne z$, the effective resistance $R_c(a\leftrightarrow z)$ 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 $R_c(a\leftrightarrow z)=\|I\|^{-1}$.

The theorem (Rayleigh's Monotonicity Law, Theorem 9.12 of Levin–Peres–Wilmer) asserts: if $c'(x,y)\le c(x,y)$ on every edge — resistances are only increased — then

Decreasing conductances can only increase effective resistance. Deceptively simple, this is one of the most-used facts of the theory: it lets one bound resistances in a complicated network by deleting edges (setting conductances to zero) or by comparison with a tractable subnetwork, and through the commute-time identity it transfers to monotonicity statements for hitting times.