From 0c88417ea2f465e3d7add5a7e44793e59bbaf5ab Mon Sep 17 00:00:00 2001 From: Andrew Date: Sun, 23 Feb 2025 04:56:53 +0000 Subject: [PATCH] [CS4423]: WK06 --- .../CS4423/materials/CS4423-W06-Part-1.pdf | Bin 0 -> 238689 bytes .../CS4423/materials/CS4423-W06-Part-2.pdf | Bin 0 -> 154183 bytes .../CS4423/materials/CS4423-W06-Part-3.pdf | Bin 0 -> 176765 bytes year4/semester2/CS4423/notes/CS4423.pdf | Bin 1737748 -> 1744594 bytes year4/semester2/CS4423/notes/CS4423.tex | 59 ++++++++++++++++++ 5 files changed, 59 insertions(+) create mode 100644 year4/semester2/CS4423/materials/CS4423-W06-Part-1.pdf create mode 100644 year4/semester2/CS4423/materials/CS4423-W06-Part-2.pdf create mode 100644 year4/semester2/CS4423/materials/CS4423-W06-Part-3.pdf diff --git a/year4/semester2/CS4423/materials/CS4423-W06-Part-1.pdf b/year4/semester2/CS4423/materials/CS4423-W06-Part-1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e1008dec7669e9f6b183d40221a591f9a01ce8b6 GIT binary 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a/year4/semester2/CS4423/notes/CS4423.tex +++ b/year4/semester2/CS4423/notes/CS4423.tex @@ -749,6 +749,65 @@ Then: \end{align*} \end{itemize} +% MISSING WEEK06-01 shit + +\section{Centrality Measures} +Key nodes in a network can be identified through \textbf{centrality measures}: a way of assigning ``scores'' to nodes that represents their ``importance''. +However, what it means to be central depends on the context; +accordingly, in the context of network analysis, a variety of different centrality measures have been developed. +Measures of centrality include: +\begin{itemize} + \item \textbf{Degree centrality:} just the degree of the nodes, important in transport networks for example. + \item \textbf{Eigenvector centrality:} defined in terms of properties of the network's adjacency matrix. + \item \textbf{Closeness centrality:} defined in terms of a node's \textbf{distance} to other nodes in the network. + \item \textbf{Betweenness centrality:} defined in terms of \textbf{shortest paths}. +\end{itemize} +\subsection{Degree Centrality} +In a (simple) graph $G=(X,E)$ with $X=\{ 0, 1, \dots, n-1 \}$ and adjacency matrix $A=(a_{i,j})$, the \textbf{degree centrality} $c_i^D$ of node $i \in X$ is defined as: +\[ + c_i^D = k_i = \sum_j a_{i,j} +\] + +where $k_i$ is the degree of node $i$. +\\\\ +In some cases, this measure can be misleading since it depends (among other things) on the order of the graph. +A better measure is the \textbf{normalised degree centrality}: the normalised degree centrality $C_I^D$ of node $i \in X$ is defined as: +\[ + C_i^D = \frac{k_i}{n-1} = \frac{c_i^D}{n-1} \left( = \frac{\text{degree of centrality of node } i}{\text{number of potential neighbours of } i} \right) +\] + +Note that in a directed graph, one distinguishes between the \textbf{in-degree} and the \textbf{out-degree}o f a node and defines the in-degree centrality and the out-degree centrality accordingly. + +\subsection{Eigenvector Centrality} +Let $A$ be a square $n \times n$ matrix. +An $n$-dimensional vector, $v$, is called an \textbf{eigenvector} of $A$ if : +\[ + Av = \lambda v +\] + +for some scalar $\lambda$ which is called an \textbf{eigenvalue} of $A$. +\\\\ +When $A$ is a real-valued matrix, one usually finds that $\lambda$ and $v$ are \textit{complex-valued}. +However, if $A$ is \textbf{symmetric}, then they are \textit{real-valued}. +$A$ may have up to $n$ eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$. +The \textbf{spectral radius} of $A$ is $\rho(A) := \text{max}(|\lambda_1|, \lambda_2|, \dots, |\lambda_n|)$. +If $v$ is an eigenvector associated with the eigenvalue $\lambda$, so too is any non-zero multiple of $v$. +\\\\ +The basic idea of eigenvector centrality is that a node's ranking in a network should relate to the rankings of the nodes it is connected to. +More specifically, up to to some scalar $\lambda$, the centrality $c_i^E$ of node $i$ should be equal to the sum of the centralities $c_j^E$ of its neighbouring nodes $j$. +In terms of the adjacency matrix $A = (a_{i,j})$, this relationship is expressed as: +\[ + \lambda c_i^E = \sum_j a_{i,j} c_j^E +\] + +which, in turn, in matrix language is: +\[ + \lambda c^E = Ac^E +\] + +for the vector $c^E = (c_i^E)$ which then is an eigenvector of $A$. +So $c^E$ is an eigenvector of $A$ (but which one?). +