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  "body": "\n## Definition\n\nLet $(G, c)$ be a finite [[electrical network|T7KRE7J3B4]] with vertex set $V$, and let $a, b \\in V$ with $a \\ne b$.\nLet $v \\colon V \\to \\R$ be the unique function with $v(a) = 1$ and $v(b) = 0$ that is [[harmonic|WRY0M7SW59]] on $V \\setminus \\set{a, b}$.\nThis is the solution of a [[discrete Dirichlet problem|YKETR75ZYW]] with boundary $\\set{a, b}$.\nLet $I = \\sum_{y} c(a, y)\\big(1 - v(y)\\big)$ be the net current that leaves $a$.\nThe *effective resistance* between $a$ and $b$ is $R_{\\mathrm{eff}}(a, b) = 1/I$.\nThe number $C_{\\mathrm{eff}}(a, b) = I$ is the *effective conductance*.\n\n## Remarks\n\nThe current $I$ is positive: $0 \\le v \\le 1$ by the maximum principle, and $v(y) \u003c 1$ for some neighbour $y$ of $a$ since $G$ is connected.\nScaling $v$ shows that $R_{\\mathrm{eff}}(a, b)$ is the voltage difference between $a$ and $b$ when a current of strength $1$ enters at $a$ and leaves at $b$.\nExample: for the path $a - m - b$ with unit conductances, $v(m) = \\tfrac12$, so $I = \\tfrac12$ and $R_{\\mathrm{eff}}(a, b) = 2$, the sum of the two resistances in series.\nIf an edge of conductance $1$ is added between $a$ and $b$, the current becomes $\\tfrac12 + 1$, so $R_{\\mathrm{eff}}(a, b) = \\tfrac23$, as for resistors in parallel.\nNon-example: if $a$ and $b$ lie in different components of a disconnected graph, no current can flow between them.\nThen the effective resistance is not a finite number, which is why connectedness is part of the definition of a network.\nWith $L$ the weighted [[graph Laplacian|TN6M6DVZRP]] and $L^{+}$ its Moore–Penrose pseudoinverse, $R_{\\mathrm{eff}}(a, b) = (e_a - e_b)^{T} L^{+} (e_a - e_b)$, where $e_x$ is the indicator vector of $x$.\nThe function $(a, b) \\mapsto R_{\\mathrm{eff}}(a, b)$, extended by $0$ on the diagonal, is a metric on $V$, the resistance distance.\n",
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