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zNLQ!?N$yk6DvV4v!JfUuS39b5yoJ_L1#@d|?jhik{`>XRPjXzWm3Bj?a%GiOT(`j9 z_cTNW1`kv!rVk1#;IH0Qfo=v{?p6Be{Edj)zIWjx-i%LDDv+SdM7yXI(~N)`4L#;# zyOx<>tnI=z?wU%g?V?7_uGS}mw z3R^fL({q)i^~MR;w_THhJoV%*sIy;v3Wld)*XFWb3^}iR#;MM)#-1&8oYu><92X`f;_bs*@WIB|d_wRUA{lnX`@UvyV@jZZke^C(56& zUoVg)z%u@P)GCd)y{2|ZYjrtqkjKmrGl6QB+Sc5al;{(%vCx7B_@}kATJ*;8W1R_ HA&>T7miBoE diff --git a/year4/semester2/CS4423/notes/CS4423.tex b/year4/semester2/CS4423/notes/CS4423.tex index e63857cb..b6c7f5b9 100644 --- a/year4/semester2/CS4423/notes/CS4423.tex +++ b/year4/semester2/CS4423/notes/CS4423.tex @@ -1142,10 +1142,10 @@ It can be helpful to think about $P_n$. \subsubsection{Characteristic Path Length} The \textbf{characteristic path length} (i.e., the average shortest path length) $L$ of a graph $G$ is the average distance between pairs of nodes: \[ - L = \frac{1}{n(n-1)} \sum_i \sum_j d_{i,j} + L = \frac{1}{n(n-1)} \sum_{i \neq j} d_{i,j} \] -For graphs drawn from $G_{ER}(n,m)$ and $G_{ER}(n,p)$, $L = \frac{\ln(n)}{\ln( \langle k \rangle)}$. +For graphs drawn from $G_{ER}(n,m)$ and $G_{ER}(n,p)$, $L = \frac{\ln(n)}{\ln( \langle k \rangle)}$, where $\langle k \rangle$ is the average degree of the network. \subsubsection{Clustering} In contrast to random graphs, real-world networks also contain \textbf{many triangles}: it is not uncommon that a friend of one of my friends is also my friend. @@ -1168,7 +1168,89 @@ This proportion can be computed as follows: where $n_\Delta$ is the number of triangles in $G$ and $n_\land$ is the number of triads. +\subsubsection{Small World Behaviour} +A network $G = (X,E)$ is said to exhibit \textbf{small world behaviour} if its characteristic path length $L$ grows proportionally to the logarithm of the number of nodes of $G$: +\[ + L \sim \ln(n) +\] +In this sense, the ensembles $G(n,m)$ \& $G(n,p)$ of random graphs do exhibit small world behaviour (as $n \to \infty$). + +\subsubsection{Transitivity} +The \textbf{transitivity} $T$ of a graph $G=(X,E)$ is the proportion of \textbf{transitive} triads, i.e., triads which are subgraphs of \textbf{triangles}. +This proportion can be computed as: +\[ + T = \frac{3n_\Delta}{n_\land} +\] + +where $n_\Delta$ is the number of triangles in $G$ and $n_\land$ is the number of triads. +\\\\ +The transitivity of a graph in $G_{ER}(n,p)$ is easy to estimate: +for every triad, the ``third'' edge is present with probability $p$, so: +\[ + T = p +\] + +Or, compute $\frac{3n_\Delta}{n_\land}$ using the explicit formulas from the previous lecture: +$n_\Delta = \binom{n}{3} p^3$ and $n_\land = 3 \binom{n}{3}p^2$. + +\subsection{Clustering} +The concept of \textbf{clustering} measures the transitivity of a node, or of an entire graph in a different way. +To define it, we need the concept of an \textbf{induced subgraph}. + +\subsubsection{Induced Subgraph} +Given $G = (X,E)$ and $Y \subset X$, the \textbf{induced subgraph} of $G$ on $Y$ is the graph $H = \left( Y, E \cap \binom{Y}{2} \right)$. +That is: +\begin{itemize} + \item $H$ is a subgraph of $G$ with node set $Y$. + \item $H$ has all possible edges in $G$ for which both nodes are in $Y$. +\end{itemize} + +\subsubsection{Clustering Coefficient} +For a node $i \in X$ of a graph $G = (X,E)$, denote by $G_i$ the subgraph induced on the neighbours of $i$ in $G$, and by $m(G_i)$ its number of edges; +the \textbf{node clustering coefficient} $c_i$ of node $i$ is defined as: +\[ + c_i = + \begin{cases} + \binom{k_i}{2}^{-1} m(G_i) & k_i \geq 2 \\ + 0 & \text{otherwise} + \end{cases} +\] + +That is, the node clustering coefficient measures the proportion of existing edges in its \textbf{social graph} among the possible edges. +\\\\ +The \textbf{graph clustering coefficient} $C$ of $G$ is the average node clustering coefficient: +\[ + C = \langle c \rangle c = \frac{1}{n} \sum^n_{i=1} c_i +\] + +By definition, $0 \leq c_i \leq 1$ for all nodes $i \in X$, and $0 \leq C \leq 1$. +\\\\\ +The \textbf{node clustering coefficient} of any node $i$ in a $G_{ER}(n,p)$ \textbf{random graph} is $c_i = p$, i.e., in any selection of potential edges, by construction a proportion $p$ of them is present in the random graph; +this is true in particular for the $\binom{k}{2}$ potential edges between the $k$ neighbours of a node of degree $k$. +The \textbf{graph clustering coefficient} of a $G_{ER}(n,p)$ \textbf{random graph} is: +\[ + C = p +\] + +Note that when $p(n) = \langle k \rangle n^{-1}$ for a fixed expected average degree $\langle k \rangle$, then $C = \frac{\langle k \rangle}{n} \to 0$ for $n \to \infty$; +that is, in large $G_{ER}$ random graphs, the number of triangles is negligible. +In real-world networks, one often observers that $\frac{C}{\langle k \rangle}$ does not depend on $n$ (as $n \to \infty$). + +\subsubsection{Clustering versus Transitivity} +For a node $i \in X$, denote by $n^\land_i = \binom{k_i}{2}$ the number of triads containing $i$ as their central node, and by $n_i^\Delta$ the actual number of triangles containing $i$; +then, the node clustering coefficient is: +\begin{align*} + c_i = \frac{n_i^\Delta}{n_i^\land} \quad \text{ or,} \\ + n_i^\Delta = n_i^\land c_i +\end{align*} + +Moreover, $3n_\Delta = \sum_i n_i^\Delta$ and $n_\land = \sum_i n_i^\land$; +it follows that $T = \frac{3n_\Delta}{n_\land} = \frac{1}{n_\land} \sum_i n_i^\land c_i$, in contrast to $C = \frac{1}{n} \sum_i c_i$. +That is, $C$ is the (plain) \textbf{average} of the node clustering coefficients, whereas $T$ is a \textbf{weighted average} of node clustering coefficients, giving higher weight to high-degree nodes. +\\\\ +The fact that ER random networks tend to have low transitivity \& clustering shows the need for a new kind of (random) network construction that is better at modelling real-world networks. +One idea is to start with some \textbf{regular network} that naturally has \textit{high clustering}, and then to randomly distort its edges to introduce some \textbf{short paths}.