Random graph

Remarks

In $G(n, p)$ 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. Example: $G(n, 1/2)$ gives each of the $2^{\binom{n}{2}}$ graphs in $\mathcal{G}_n$ the same probability, so it is the uniform random graph on $V$. Non-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 $1$. A property holds asymptotically almost surely in a model if its probability tends to $1$ as $n \to \infty$. Another standard model is $G(n, M)$, the uniform measure on the graphs in $\mathcal{G}_n$ with exactly $M$ edges.