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  "body": "\n## Summary\n\nIn discrete potential theory, an electrical network is a graph whose edges carry positive conductances, the reciprocals of resistances.\nVoltages on the vertices and currents on the edges obey Ohm's law and Kirchhoff's laws.\nA voltage with no sources at a vertex is a [[discrete harmonic function|WRY0M7SW59]] there.\nNetworks turn questions about [[random walks|04VZE3FZXF]], such as recurrence, into questions about [[effective resistance|XAE5YJ3H68]].\n\n## Explanations\n\n### L1 human\n\nPicture a graph made of wires, where each wire has a resistance.\nConnect a battery to two vertices; a current flows and every vertex gets a voltage.\nAt every other vertex the current in equals the current out, and this makes the voltage there a weighted average of the neighbouring voltages.\nFor two unit resistors in a row between $a$ and $b$, the voltage at the middle vertex is halfway between the voltages at $a$ and $b$.\nThe same averaging rule governs a random walker on the graph, which is why circuits help to study random walks.\n",
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