Introduction to the Modeling and Analysis of Complex Systems

378CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESnx.draw(g, positions, with_labels = True)show()The result is shown in Fig. 17.4.2492825 27231231 22231372933328120 2130140191726 1815111046 516Figure 17.4: Visual output of Code 17.7.We can use this shortest path length to define several useful metrics to characterizethe network’s topological properties. Let’s denote a shortest path length from node i tonode j as d(i → j). Clearly, d(i → i) = 0 for all i. Then we can construct the followingmetrics:Characteristic path length∑i,jd(i → j)L =n(n − 1)(17.15)where n is the number of nodes. This formula works for both undirected anddirected networks. It calculates the average length of shortest paths for allpossible node pairs in the network, giving an expected distance between tworandomly chosen nodes. This is an intuitive characterization of how big (orsmall) the world represented by the network is.