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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $V = \\set{1, \\dots, n}$, and let $\\mathcal{G}_n$ be the finite set of all simple [[graphs|0JD5C82QMB]] with vertex set $V$.\nA random graph on $V$ is a [[probability measure|0VGMEBVW1M]] $\\mathbb{P}$ on $\\mathcal{G}_n$.\nThat is, it is a map $\\mathbb{P} \\colon \\mathcal{G}_n \\to [0, 1]$ with $\\sum_{G \\in \\mathcal{G}_n} \\mathbb{P}(G) = 1$.\nEquivalently, it is the law of a random variable on a [[probability space|0ZYBZ34RT2]] with values in $\\mathcal{G}_n$.\nA model of random graphs is a choice of such a measure $\\mathbb{P}_n$ for every $n$.\nFor $0 \\le p \\le 1$, the binomial model $G(n, p)$ is the measure with $\\mathbb{P}(G) = p^{m} (1 - p)^{\\binom{n}{2} - m}$, where $m$ is the number of edges of $G$.\n\n## Remarks\n\nIn $G(n, p)$ the $\\binom{n}{2}$ possible edges are present independently, each with probability $p$; the probabilities add up to $(p + (1 - p))^{\\binom{n}{2}} = 1$ by the binomial theorem.\nExample: $G(n, 1/2)$ gives each of the $2^{\\binom{n}{2}}$ graphs in $\\mathcal{G}_n$ the same probability, so it is the uniform random graph on $V$.\nNon-example: giving each of the $8$ graphs on $\\set{1, 2, 3}$ the weight $1/2$ is not a random graph, because the weights add up to $4$, not $1$.\nA property holds asymptotically almost surely in a model if its probability tends to $1$ as $n \\to \\infty$.\nAnother standard model is $G(n, M)$, the uniform measure on the graphs in $\\mathcal{G}_n$ with exactly $M$ edges.\n",
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