Introduction to the Modeling and Analysis of Complex Systems

17.3. CENTRALITIES AND CORENESS 383disseminate a rumor to the entire network, closeness centrality would probably be a moreappropriate metric to use. Or, if you want to find the most effective person to monitor andmanipulate information flowing within the network, betweenness centrality would be moreappropriate, assuming that information travels through the shortest paths between people.Eigenvector centrality and PageRank are useful for generating a reasonable rankingof nodes in a complex network made of directed edges.Exercise 17.5 Visualize the Karate Club graph using each of the above centralitymeasures to color the nodes, and then compare the visualizations to see howthose centralities are correlated.Exercise 17.6 Generate (1) an Erdős-Rényi random network, (2) a Watts-Strogatz small-world network, and (3) a Barabási-Albert scale-free network of comparablesize and density, and obtain the distribution of node centralities of yourchoice for each network. Then compare those centrality distributions to find whichone is most/least heterogeneous.Exercise 17.7 Prove that PageRank with α = 1 is essentially the same as thedegree centrality for undirected networks.A little different approach to characterize the centrality of nodes is to calculate theircoreness. This can be achieved by the following simple algorithm:1. Let k = 0.2. Repeatedly delete all nodes whose degree is k or less, until no such nodes exist.Those removed nodes are given a coreness k.3. If there are still nodes remaining in the network, increase k by 1, and then go backto the previous step.Figure 17.5 shows an example of this calculation. At the very end of the process, we seea cluster of nodes whose degrees are at least the final value of k. This cluster is called ak-core, which can be considered the central part of the network.In NetworkX, you can calculate the corenesses of nodes using the core_number function.Also, the k-core of the network can be obtained by using the k_core function. Hereis an example: