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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $\\mathcal{G}_n$ be the set of all simple [[graphs|00C0BR7D7H]] with vertex set $[n] = \\set{1, \\ldots, n}$.\nLet $N = \\binom{n}{2}$ be the number of pairs of distinct vertices; see [[binomial coefficient|0GBZM0MT0F]].\n\n1. For $0 \\le p \\le 1$, the model $G(n, p)$ is the [[probability measure|059TJN0NZ1]] on $\\mathcal{G}_n$ giving each graph with $m$ edges the probability $p^m (1 - p)^{N - m}$.\n2. For an integer $M$ with $0 \\le M \\le N$, the model $G(n, M)$ is the uniform probability measure on the set of graphs in $\\mathcal{G}_n$ with exactly $M$ edges.\n\n## Remarks\n\nIn $G(n, p)$ the $N$ events \"the pair $\\set{i, j}$ is an edge\" are independent, each of probability $p$.\nSo the number of edges has the binomial distribution with parameters $N$ and $p$, and its mean is $pN$.\n\nExample: for $n = 3$ and $p = 1/2$, every graph on $\\set{1, 2, 3}$ has probability $(1/2)^3 = 1/8$, and these $8$ values sum to $1$.\n\nNon-example: on $\\set{1, 2, 3}$, choosing the triangle with probability $1/2$ and the empty graph with probability $1/2$ is not $G(3, p)$ for any $p$.\nIts edge probability is $1/2$, but the three edge events are not independent, since the graph with one edge has probability $0$.\n\nA random graph from these models is also written $G(n, p)$ or $G(n, M)$.\nFor $M$ near $pN$ the two models behave in a similar way.\n",
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