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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a [[graph|00C0BR7D7H]] in which every vertex has finitely many neighbours.\nLet $c \\colon V \\times V \\to [0, \\infty)$ be a symmetric weight with $c(x, y) \u003e 0$ if and only if $\\set{x, y} \\in E$, and let $c(x) = \\sum_{y} c(x, y)$.\nA function $f \\colon V \\to \\R$ is *harmonic at* a vertex $x$ with $c(x) \u003e 0$ if\n$$f(x) = \\frac{1}{c(x)} \\sum_{y \\in V} c(x, y) f(y),$$\nor equivalently if $\\sum_{y \\in V} c(x, y) \\big(f(y) - f(x)\\big) = 0$.\nFor $A \\subseteq V$, the function $f$ is *harmonic on* $A$ if it is harmonic at every vertex of $A$.\nWith unit weights on all edges, harmonicity at $x$ says that $f(x)$ is the average of $f$ over the neighbours of $x$.\n\n## Remarks\n\nExample: on the graph $\\Z$ with edges $\\set{n, n + 1}$ and unit weights, $f(n) = 3n + 2$ is harmonic on $\\Z$, since $\\tfrac12\\big(f(n - 1) + f(n + 1)\\big) = \\tfrac12(6n + 4) = f(n)$.\nNon-example: $g(n) = n^2$ is harmonic at no vertex, since $\\tfrac12\\big((n - 1)^2 + (n + 1)^2\\big) = n^2 + 1$.\nThe expression $\\sum_y c(x, y)(f(y) - f(x))$ is minus the weighted [[graph Laplacian|TN6M6DVZRP]] applied to $f$ at $x$, so the harmonic functions on $V$ form the kernel of the Laplacian.\nFor the random walk with transition probabilities $p(x, y) = c(x, y)/c(x)$, the function $f$ is harmonic at $x$ exactly when $f(x)$ is the expected value of $f$ after one step from $x$.\nThis is the discrete form of the mean value property of a [[harmonic function|0DEB0JMN0W]].\n",
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