---
tag: KX4152NVW8
kind: statement
role: theorem
name: Rayleigh monotonicity law
msc: [31C20, 94C05, 60J10]
uses: [T7KRE7J3B4, XAE5YJ3H68, 8BPMHA5MT4]
about: [XAE5YJ3H68]
sources:
  - url: https://en.wikipedia.org/wiki/Resistance_distance
---

## Statement

Let $G = (V, E)$ be a finite connected graph, and let $r, r' \colon E \to (0, \infty)$ be two resistance functions.
So $(G, 1/r)$ and $(G, 1/r')$ are [[electrical networks|T7KRE7J3B4]].
Suppose that $r(e) \le r'(e)$ for every edge $e \in E$.
Then for all vertices $a \ne b$,
$$R_{\mathrm{eff}}(a, b) \le R'_{\mathrm{eff}}(a, b),$$
where $R_{\mathrm{eff}}$ and $R'_{\mathrm{eff}}$ are the [[effective resistances|XAE5YJ3H68]] for $r$ and for $r'$.

## Remarks

By [[Thomson's principle|8BPMHA5MT4]], each effective resistance is the minimum energy of a unit flow from $a$ to $b$.
For every unit flow $\theta$, the energy $\sum_e r(e) \theta(e)^2$ is at most $\sum_e r'(e) \theta(e)^2$.
So the minimum for $r$ is at most the minimum for $r'$.
Example: the path $a - m - b$ with unit resistances has $R_{\mathrm{eff}}(a, b) = 2$.
Adding an edge from $a$ to $b$ of resistance $1$ gives $R_{\mathrm{eff}}(a, b) = \tfrac23$.
This agrees with the law, viewing the new edge as the limit of an edge of very large resistance whose resistance is lowered.
Two consequences are standard: removing an edge, as long as the graph stays connected, does not decrease effective resistances.
Gluing two vertices into one, which amounts to giving a resistance $0$ to an edge between them, does not increase effective resistances.

## Explanations

### L1 human

If some wires in a network get worse, the network as a whole cannot get better.
Raising the resistance of any wire can only raise the resistance between any two points.
Cutting a wire is an extreme case of raising its resistance, and joining two points by a perfect wire is an extreme case of lowering it.
The law sounds obvious, but its proof needs a variational principle such as Thomson's principle.

### L2 human

The law is the main tool for comparing random walks on different graphs.
To show that a walk on a graph is recurrent, one may glue vertices together, which lowers resistances, and show that the resistance to infinity is still infinite.
To show transience, one may delete edges, which raises resistances, and find a subgraph, such as a tree, with finite resistance to infinity.
Combined with the criterion that relates recurrence to the resistance to infinity, this proves the transience of $\Z^3$ from the transience of a suitable subgraph.