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+then, the average degree $\langle k \rangle = pn = c$ remains fixed as $n \to \infty$. +For graphs $G_{ER}(n,p)$: +\begin{itemize} + \item If $c < 1$, the graph contains many small components with orders bounded by $O(\ln(n))$. + \item If $c=1$ the graph has large components of order $S = O(n^\frac{2}{3})$. + \item If $c > 1$, there is a unique \textbf{giant component} of order $S = O(n)$. +\end{itemize} +\subsection{Small World Network} +Many real-world networks are \textbf{small world networks}, wherein most pairs of nodes are only a few steps away from each other, and where nodes to form \textit{cliques}, i.e., subgraphs in which all nodes are connected to each other. +Three network attributes that measure these small-world effects are: +\begin{itemize} + \item \textbf{Characteristic path length}, $L$: the average length of all shortest paths in the network. + \item \textbf{Transitivity}, $T$: the proportion of \textit{triads} that form triangles. + \item \textbf{Clustering coefficient}, $C$: the average node clustering coefficient. +\end{itemize} + +A network is called a \textbf{small world network} if it has: +\begin{itemize} + \item A small \textbf{average shortest path length} $L$ (scaling with $\log(n)$, where $n$ is the number of nodes) and + \item A high \textbf{clustering coefficient} $C$. +\end{itemize} + +It turns out that ER random networks do have a small average shortest path length, but not a high clustering coefficient. +This observation justifies the need for a different model of random networks, if they are to be used to model the clustering behaviour of real-world networks. + +\subsubsection{Distance} +We have seen how BFS can determine the length of a shortest path from a given node $x$ to any node $y$ in a \textit{connected network}. +An application to all nodes $x$ yields the shortest distances between all pairs of nodes. +Recall that the \textbf{distance matrix} of a connected graph $G = (X,E)$ is $\mathcal{D} = (d_{i,j})$ where entry $d_{i,j}$ is the length of the shortest path from node $i \in X$ to node $j \in X$. +(Note that $d_{i,i} = 0$ for all $i$). +There are a number of graph (and node) attributes that can be defined in terms of this matrix: +\begin{itemize} + \item The \textbf{eccentricity} $e_i$ of a node $i \in X$ is the maximum distance between $i$ and any other vertex in $G$, so $e_i = \text{max}_j(d_{i,j})$. + \item The \textbf{graph radius} $R$ is the minimum eccentricity, $R = \text{min}_i(e_i)$. + \item The \textbf{graph diameter} $D$ is the maximum eccentricity: $D = \text{max}_i(e_i) = - \text{max}_{i,j} (d_{i,j})$. +\end{itemize} + +Note that one shouldn't think that the ``diameter is twice the radius'', but rather diameter is the distance between the points furthest from each other and radius is the distance from the ``centre'' to the furthest point from it. +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: +\[ + L = \frac{1}{n(n-1)} \sum_i \sum_j d_{i,j} +\] + +For graphs drawn from $G_{ER}(n,m)$ and $G_{ER}(n,p)$, $L = \frac{\ln(n)}{\ln( \langle k \rangle)}$. + +\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. +This \textbf{degree of transitivity} can be measured in several different ways. +For the first, we need two concepts: +\begin{itemize} + \item The number of \textbf{triangles} in $G$, denoted $n_\Delta$, is the number of subgraphs of $G$ that are isomorphic to $C_3$. + \item The number of \textbf{triads} in $G$, denoted $n_\land$, is the number of pairs of edges with a shared node. +\end{itemize} + +There is an easy way to count the number of \textbf{triads} in a network: +if node $i$ has degree $k_i = \text{deg}(i)$, then it is involved in $\binom{k_i}{2}$ triads, +so the total number of triads is $n_\land = \sum_i \binom{k_i}{2}$. +\\\\ +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 follows: +\[ + T = 3 \frac{n_\Delta}{n_\land} +\] + +where $n_\Delta$ is the number of triangles in $G$ and $n_\land$ is the number of triads.