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standard deviation $\sigma_k = \sqrt{p(1-p) (n-1)}$. + + +\subsubsection{Degree Distribution} +The \textbf{degree distribution} $p: \mathbb{N}_0 \to \mathbb{R}, k \mapsto p_k$ of a graph $G$ is defined as +\begin{align*} + p_k = \frac{n_k}{n} +\end{align*} +where, for $k \geq 0$, $n_k$ is the number of nodes of degree $k$ in $G$. +This definition can be extended to ensembles of graphs with $n$ nodes (like the random graphs $G(n,m)$ and $G(n,p)$) by setting +\begin{align*} + p_k \frac{\bar{n}_k}{n} +\end{align*} + +where $\bar{n}_k$ denotes the expected value of the random graph $n_k$ over the ensemble of graphs. +\\\\ +The degree distribution in a random graph $G(n,p)$ is a \textbf{binomial distribution}: +\begin{align*} + p_k = \binom{n-1}{k}p^k (1-p)^{n-1-k} = \text{bin}(n-1,p,k) +\end{align*} + +That is, in the $G(n,p)$ model, the probability that a nodes has degree $k$ is $p_k$. +Also, the \textbf{average degree} of a randomly chosen node is +\begin{align*} + \langle k \rangle = \sum^{n-1}_{k=0} kp_k = p(n-1) +\end{align*} + +(with standard deviation $\sigma_k = \sqrt{p(1-p)(n-1))}. + + diff --git a/year4/semester2/CS4423/notes/images/gnm.png b/year4/semester2/CS4423/notes/images/gnm.png new file mode 100644 index 0000000000000000000000000000000000000000..d590aa421efdd5e5e9149cff9da9ac1ba6050d97 GIT binary patch literal 224682 zcmdSA^;?u%+&9Wr1Vlrt8fm3r=VYMlu=M# zpvb+E(D2UMnbmPN@Clqe-@mu_LT5WmL6viMI%`ds<$iVQtw9?nn#!uRj1Etl2l z(y*j)l-S(=PR*s^C_Yl%qAPXvkTI1tj+aemSetwJhi`^q@zjo^;{s;Ud1uxlCiF4}P77 z@SVrqZK;itA$y`;x$3DTO7a7hHwoQUSdHAAvDmeT17;6Jct>Mj>3sQjQxMCdt;1Vo zs|)A+pZ%^W)np&;OmT^1ND&(D#pF2Wb%!w}SKc;-N$C!bXT#vl;p)0}EmKi=lHm-q zVaq;uQwXgLZDuo4OgI;3a$}W=Mmokzfl=v4Wl;%r7WG1C!}(2XrwEf>~y)&HXHrkx|3$;!6oO)s2E%K2pcYP~PNiG3tYDhO+}Qqp7H>!HFlxA`5@J zTyQ9IzaGcNnNcxwO^NfhF@|z4==PHOZjS8QItmfzHYJI1_$J{>7MvSh!ni+2&U$XE zN60s#;SoYO4|cNJwqyFo@p+Qa{8yXCWN2--cc?)Jn>;3TLskDSn35ODYgL?fIpEcu zVNm`zC1UoPQi*BU1yb*DfIQOMB=rlElXRnxc#UO_@V;P~Q< 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