Introduction to the Modeling and Analysis of Complex Systems

382CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESthat reach node i in t steps, with t taken to infinity. Eigenvector v is usuallychosen to be a non-negative unit vector (v i ≥ 0, |v| = 1).PageRankc P (i) = v i(i-th element of the dominant eigenvector v of thefollowing transition probability matrix) (17.23)T = αAD −1 + (1 − α) J n(17.24)where A is the adjacency matrix of the network, D −1 is a diagonal matrix whosei-th diagonal component is 1/deg(i), J is an n × n all-one matrix, and α is thedamping parameter (α = 0.85 is commonly used by default).PageRank is a variation of eigenvector centrality that was originally developedby Larry Page and Sergey Brin, the founders of Google, in the late 1990s [74,75] to rank web pages. PageRank measures the asymptotic probability for arandom walker on the network to be standing on node i, assuming that thewalker moves to a randomly chosen neighbor with probability α, or jumps toany node in the network with probability 1 − α, in each time step. Eigenvector vis usually chosen to be a probability distribution, i.e., ∑ i v i = 1.Note that all of these centrality measures give a normalized value between 0 and 1,where 1 means perfectly central while 0 means completely peripheral. In most cases, thevalues are somewhere in between. Functions to calculate these centrality measures arealso available in NetworkX (outputs are omitted in the code below to save space):Code 17.9:>>> import networkx as nx>>> g = nx.karate_club_graph()>>> nx.degree_centrality(g)>>> nx.betweenness_centrality(g)>>> nx.closeness_centrality(g)>>> nx.eigenvector_centrality(g)>>> nx.pagerank(g)While those centrality measures are often correlated, they capture different propertiesof each node. Which centrality should be used depends on the problem you are addressing.For example, if you just want to find the most popular person in a social network,you can just use degree centrality. But if you want to find the most efficient person to