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  "body": "\n## Statement\n\nLet $G = (V, E)$ be a finite connected graph, and let $r, r' \\colon E \\to (0, \\infty)$ be two resistance functions.\nSo $(G, 1/r)$ and $(G, 1/r')$ are [[electrical networks|T7KRE7J3B4]].\nSuppose that $r(e) \\le r'(e)$ for every edge $e \\in E$.\nThen for all vertices $a \\ne b$,\n$$R_{\\mathrm{eff}}(a, b) \\le R'_{\\mathrm{eff}}(a, b),$$\nwhere $R_{\\mathrm{eff}}$ and $R'_{\\mathrm{eff}}$ are the [[effective resistances|XAE5YJ3H68]] for $r$ and for $r'$.\n\n## Remarks\n\nBy [[Thomson's principle|8BPMHA5MT4]], each effective resistance is the minimum energy of a unit flow from $a$ to $b$.\nFor every unit flow $\\theta$, the energy $\\sum_e r(e) \\theta(e)^2$ is at most $\\sum_e r'(e) \\theta(e)^2$.\nSo the minimum for $r$ is at most the minimum for $r'$.\nExample: the path $a - m - b$ with unit resistances has $R_{\\mathrm{eff}}(a, b) = 2$.\nAdding an edge from $a$ to $b$ of resistance $1$ gives $R_{\\mathrm{eff}}(a, b) = \\tfrac23$.\nThis agrees with the law, viewing the new edge as the limit of an edge of very large resistance whose resistance is lowered.\nTwo consequences are standard: removing an edge, as long as the graph stays connected, does not decrease effective resistances.\nGluing two vertices into one, which amounts to giving a resistance $0$ to an edge between them, does not increase effective resistances.\n\n## Explanations\n\n### L1 human\n\nIf some wires in a network get worse, the network as a whole cannot get better.\nRaising the resistance of any wire can only raise the resistance between any two points.\nCutting 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.\nThe law sounds obvious, but its proof needs a variational principle such as Thomson's principle.\n\n### L2 human\n\nThe law is the main tool for comparing random walks on different graphs.\nTo 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.\nTo show transience, one may delete edges, which raises resistances, and find a subgraph, such as a tree, with finite resistance to infinity.\nCombined 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.\n",
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