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  "body": "\n## Summary\n\nThe Erdős–Rényi model is the simplest model of a [[random graph|TNT3DG0HRJ]].\nIn the form $G(n, p)$, each of the $\\binom{n}{2}$ possible edges on $n$ labelled vertices is present independently with probability $p$.\nIn the form $G(n, M)$, a graph with exactly $M$ edges is chosen uniformly at random.\nThe model shows sharp thresholds, such as the sudden appearance of a giant component, and it is a basic tool of the [[probabilistic method|3B55HNS754]].\n\n## Explanations\n\n### L1 human\n\nTake $n$ dots and, for every pair of dots, flip a biased coin that shows heads with probability $p$.\nDraw a line between the two dots exactly when the coin shows heads.\nThe picture you get is a random graph from $G(n, p)$.\nFor example, with $n = 3$ and $p = 1/2$, each of the $8$ graphs on the vertices $1, 2, 3$ appears with probability $1/8$.\nWhen $p$ grows, the graph changes quickly from a few scattered edges to one large connected piece.\n",
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