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zgBHB4cqfo$uncb--f?6rXv5o%cMRDMQoOx*M}LtSsNfyIJA&*0op=ZFrjX0Q3cN#j zlgO1|72XNF!^kesjdv375V8lX#yf>~5ZMd*@Xp~KK=y+HJc?Z1j~oPR@Thrp0=X8f z!+Qa5A94t+$D7C7i`)P<;=PDBjvNM?@D}hqax>V1cNuRFas*Hc@9=gb#{kFf4R056 z8-L)4eF1MLatGLnm*7e2;@x=lc-xWsBagQPPq)%TQXan)Z!1zSIDyxQr^nFYP%>Y} z8%641DUV-?r&mx>Ch=C|@s^1SdIWC}Pp_|LIEJ?dj}uDNNR-yE#naKL2`QuJ!K<8Z zq6R*LH;$)cKMN?WSHDo1WEv>C$MICMS${BxryioJ$$3C&y?RLHo(GiCpT(mik_B)9 zTm+Y?U`%}J2FVM83bhQ-Ysn?cw@hlI01{9HwV(v*Ks``G$z@P3Nn=nN$zV_hNnTL$ z$Xig}NLf(I$W~CcNK{aw$Wc&^NKa6D$V^aXNJ>yr$VX5K9C@{@Mk zK(i&Uk4v@y`p>so@$Tbz-Qyu=&H2n(%iMik(gE77;nzvY6l9=c&2BzkH-B~0S@~JJ zb~zH?X)VW}NG=B}to(G7v$mll^7=rRmH+x&Qm@qmdaZ5neP?Owf;NUhzoqFT$pNqq z41x849`YN(VX(>4fp2v2X0QeD1@oGde51UkBVQq}nYaUJ67B+;e|vx?+&-YWwjXGk z@x}3)FO!yz4(s2hfacXvpnoZ)(Q?|-18?XOO&^UBRbRt{4}sUnVAT5y;G(7H?!+^g zZj8A6fqtu@@2v9nYYK!$Mh(bWdTk_LckOn>&0n;uDX+0!wDjFivy$EE&F(*R+(iFPiR%KN-@)_!zw-hf5AGfGNDRWms{nPl2+8bs&4@2 wS5>0)Io9`AAKY0mXO$aQ;|bmV+x-WYnp&Ti{>d613^X$^IW`I 0$ for all $i$. +We write $v \geq 0$. +\\\\ +\textbf{Theorem:} suppose that $A$ is a square, non-negative, \textbf{irreducible} matrix. +Then: +\begin{itemize} + \item $A$ has a real eigenvalue $\lambda = \rho(A)$ and $\lambda > |\lambda'|$ for any other eigenvalue $\lambda'$ of $A$. + $\lambda$ is called the \textbf{Perron root} of $A$. + + \item $\lambda$ is a simple root of the characteristic polynomial of $A$ (so has just one corresponding eigenvector). + + \item There is an eigenvector, $v$, associated with $\lambda$ such that $v >0$. +\end{itemize} + +For us, this means: +\begin{itemize} + \item The adjacency matrix of a connected graph has an eigenvalue that is positive and greater in magnitude than any other. + \item It has an eigenvector $v$ that is positive. + \item $v_i$ is the eigenvector centrality of the node $i$. +\end{itemize} + + +\subsection{Closeness Centrality} +A node $x$ in a network can be regarded as being central if it is \textbf{close} to (many) other nodes, as it can quickly interact with them. +Recalling that $d(i,j)$ is the distance between nodes $i$ and $j$ (i.e., the length of the shortest path between them). +Then, we can use $\frac{1}{d(i,j)}$ as a measure of ``closeness''; +in a simple, \textit{connected} graph $G=(X,E)$ of order $n$, the \textbf{closeness centrality}, $c^C_i$ of node $i$ is defined as: +\[ + c_i^C = \frac{1}{\sum_{j \in X} d(i,j)} = \frac{1}{s(i)} +\] + +where $s(i)$ is the \textbf{distance sum} for node $i$. +As is usually the case, there is a \textbf{normalised} version of this measure; +the \textbf{normalised closeness centrality} is defined as: +\[ + C_i^C = (n-1)c_i^C = \frac{n-1}{\sum_{j \in X} d(i,j)} = \frac{n-1}{s(i)} +\] + +Note that $0 \leq C_i^C \leq 1$. +\\\\ +The \textbf{distance matrix} of a graph, $G$, of order $n$ is the $n \times n$ matrix $D=(d_{i,j})$ such that: +\[ + d_{i,j} = d(i,j) +\] + +We'll return to how to compute $D$ later, but for now we note: +\begin{itemize} + \item $s(i)$ is the sum of row $i$ of $D$. + \item If $s$ is the vector of distance sums, then $s = De$ where $e = (1,1, \dots, 1)^T$. +\end{itemize} + +\subsection{Betweenness Centrality} +In a simple, connected graph $G$, the \textbf{betweenness centrality} $c_i^B$ of node $i$ is defined as: +\[ + c_i^B = \sum_j \sum_k \frac{n_i(j,k)}{n(j,k)}, \quad j \neq k \neq 1 +\] + +where $n(j,k)$ denotes the \textit{number} of shortest paths from node $j$ to node $k$, where $n_i(j,k)$ denotes the number of those shortest paths \textit{passing through} node $i$. +\\\\ +In a simple, connected graph $G$, the \textbf{normalised betweenness centrality} $c_i^B$ of node $i$ is defined as: +\[ + C_i^B = \frac{c_i^B}{(n-1)(n-2)} +\] + +