Introduction to the Modeling and Analysis of Complex Systems

386CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESCalculate the coreness of all of its nodes and draw their histogram. Compare thespeed of calculation with, say, the calculation of betweenness centrality. Also visualizethe k-core of the network.17.4 ClusteringEccentricity, centralities, and coreness introduced above all depend on the whole networktopology (except for degree centrality). In this sense, they capture some macroscopicaspects of the network, even though we are calculating those metrics for each node. Incontrast, there are other kinds of metrics that only capture local topological properties.This includes metrics of clustering, i.e., how densely connected the nodes are to eachother in a localized area in a network. There are two widely used metrics for this:Clustering coefficient∣∣ { {j, k} | d(i, j) = d(i, k) = d(j, k) = 1 } C(i) =deg(i)(deg(i) − 1)/2(17.25)The denominator is the total number of possible node pairs within node i’sneighborhood, while the numerator is the number of actually connected nodepairs among them. Therefore, the clustering coefficient of node i calculates theprobability for its neighbors to be each other’s neighbors as well. Note that thismetric assumes that the network is undirected. The following average clusteringcoefficient is often used to measure the level of clustering in the entire network:C =∑i C(i)n(17.26)TransitivityC T =∣ { (i, j, k) | d(i, j) = d(i, k) = d(j, k) = 1 }∣ ∣∣ { (i, j, k) | d(i, j) = d(i, k) = 1 } ∣ ∣(17.27)This is very similar to clustering coefficients, but it is defined by counting connectednode triplets over the entire network. The denominator is the numberof connected node triplets (i.e., a node, i, and two of its neighbors, j and k),