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  "body": "\n## Summary\n\nThe percolation threshold of a graph, also called its critical probability, is a value $p_c$ of the parameter $p$ of [[percolation|BH7YPXCYDK]].\nAt $p_c$ an infinite open cluster starts to appear.\nBelow $p_c$ all open clusters are finite almost surely, and above $p_c$ there is an infinite open cluster almost surely.\nIt depends on the graph and on the choice of bond or site percolation.\nExact values are known only for a few graphs, such as $p_c = 1/2$ for bond percolation on the square lattice.\n\n## Explanations\n\n### L1 human\n\nOpen each edge of a large network with probability $p$.\nFor small $p$ the open pieces are small islands; for large $p$ they join into one huge network.\nThe threshold is the value of $p$ at which this switch happens.\nOn a tree in which every vertex has $3$ neighbours, the threshold is $1/2$.\nEach step away from the root offers $2$ new edges, so the cluster can grow forever only if $2p$, the mean number of new open edges, exceeds $1$.\n",
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