From 95fad9e57b00bb79fb5f606bf254c97dedf44a3a Mon Sep 17 00:00:00 2001 From: Andrew Date: Wed, 5 Mar 2025 13:17:25 +0000 Subject: [PATCH] [CS4423]: WK08-1 lecture notes & materials --- year4/semester2/CS4423/exam | 1 + .../CS4423/materials/CS4423-W08-Part-1.pdf | Bin 0 -> 714031 bytes .../CS4423/materials/problems/CS4423-PS1.pdf | Bin 0 -> 141075 bytes .../materials/problems/CS4423-PS2-soln.pdf | Bin 0 -> 206493 bytes .../CS4423/materials/problems/CS4423-PS2.pdf | Bin 0 -> 150640 bytes year4/semester2/CS4423/notes/CS4423.pdf | Bin 1755688 -> 1763603 bytes year4/semester2/CS4423/notes/CS4423.tex | 50 ++++++++++++++++++ 7 files changed, 51 insertions(+) create mode 100644 year4/semester2/CS4423/exam create mode 100644 year4/semester2/CS4423/materials/CS4423-W08-Part-1.pdf create mode 100644 year4/semester2/CS4423/materials/problems/CS4423-PS1.pdf create mode 100644 year4/semester2/CS4423/materials/problems/CS4423-PS2-soln.pdf create mode 100644 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betweenness centrality} C_i^B = \frac{c_i^B}{(n-1)(n-2)} \] +\section{Random Graphs} +A \textbf{random graph} is mathematical model of a family of networks, where certain parameters (like the number of nodes \& edges) have fixed values, but other aspects (like the actual edges) are randomly assigned. +Although a random graph is not a specific object, many of its properties can be described precisely in the form of expected values or probability distributions. + +\subsection{Random Samples} +Suppose our network $G = (X,E)$ has $|X| = n$ nodes. +Then, we know that the greatest number of edges it can have is: +\begin{align*} + \binom{n}{2} = \frac{n!}{(n-2)! 2!} = \frac{n(n-1)}{2} +\end{align*} + +Our goal is to randomly select edges on the vertex set $X$, that is, pick random elements from the set $\binom{X}{2}$ of pairs of nodes. +So, we need a procedure for selecting $m$ from $N$ objects randomly, in such a way that each of the $\binom{N}{m}$ subsets of the $N$ objects is an equally likely outcome. +We first discuss sampling $m$ values in the range $\{0,1, \dots, N-1 \}$. +\begin{enumerate} + \item Suppose that we choose a natural number $N$ and a real number $p \in [0,1]$. + \item Then, iterate over each element of the set $\{0,1 \dots, N-1\}$. + \item For each, we pick a random number $x \in [0,1]$. + \item If $x < p$, we keep that number. + Otherwise, remove it from the set. +\end{enumerate} + +When we are done, how many elements do we expect in the set if $p = \frac{m}{N}$ for some chosen $m$? +And what is the likelihood of there being, say, $K$ elements in the set? +Since we are creating random samples, where the size of each is a random number, $k$, we expect that $E[k] = Np = m$; this is a \textbf{binomial distribution}: +\begin{itemize} + \item The probability of a specific subset of size $K$ to be chosen is $p^k(1-p)^{N-k}$. + \item There are $\binom{N}{k}$ subsets of size $k$, so the probability $P(k)$ of the sample to have size $k$ is $P(k) = \binom{N}{k}p^k (1-p)^{N-k}$. +\end{itemize} + +We use the following facts: +\begin{itemize} + \item $j\binom{N}{j}p^i = Np \binom{N-1}{j-1}p^{i-1}$. + \item $(1-p)^{N-j} = (1-p)^{(N-1) - (j-1)}$. + \item $(p + (1 -p))^r = 1$ for all $r$. +\end{itemize} + +The expected value is: + +\begin{align*} + E[k] =& \sum^N_{j=1}jP(j) \quad &\text{ weighted average of } j \\ + =& \sum^N_{j=0} j \binom{N}{j} p^j (1-p)^{N-j} \quad \text{ formula for } P(j) \\ + = & Np \sum^{N-1}_{l=0} \binom{N-1}{l} p^l (1-p)^{(N-1) - l} = Np \quad \text{ substituting } l=k-1 +\end{align*} + +\subsection{Erd\"os-Rényi Models} +Let $n \geq 1$, let $N = \binom{n}{2}$ and let $0 \leq m \leq N$. +The model $G_{ER}(n,m)$ consists of the ensemble of graphs $G$ on the $n$ nodes $X = \{0,1, \dots, n-1\}$, and $M$ randomly selected edges, chosen uniformly from the $N = \binom{n}{2}$ possible edges. +Equivalently, one can choose uniformly at random one network in the set $G(n,m)$ of \textit{all} networks on a given set of $n$ nodes with exactly $m$ edges. +

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