Introduction to the Modeling and Analysis of Complex Systems

17.5. DEGREE DISTRIBUTION 391er = nx.erdos_renyi_graph(n, 0.01)ws = nx.watts_strogatz_graph(n, 10, 0.01)ba = nx.barabasi_albert_graph(n, 5)Pk = [float(x) / n for x in nx.degree_histogram(er)]domain = range(len(Pk))loglog(domain, Pk, ’-’, label = ’Erdos-Renyi’)Pk = [float(x) / n for x in nx.degree_histogram(ws)]domain = range(len(Pk))loglog(domain, Pk, ’--’, label = ’Watts-Strogatz’)Pk = [float(x) / n for x in nx.degree_histogram(ba)]domain = range(len(Pk))loglog(domain, Pk, ’:’, label = ’Barabasi-Albert’)xlabel(’k’)ylabel(’P(k)’)legend()show()The result is shown in Fig. 17.9, which clearly illustrates differences between the threenetwork models used in this example. The Erdős-Rényi random network model has abell-curved degree distribution, which appears as a skewed mountain in the log-log scale(blue solid line). The Watts-Strogatz model is nearly regular, and thus it has a very sharppeak at the average degree (green dashed line; k = 10 in this case). The Barabási-Albertmodel has a power-law degree distribution, which looks like a straight line with a negativeslope in the log-log scale (red dotted line).Moreover, it is often visually more meaningful to plot not the degree distribution itselfbut its complementary cumulative distribution function (CCDF), defined as follows:∞∑F (k) = P (k ′ ) (17.29)k ′ =kThis is a probability for a node to have a degree k or higher. By definition, F (0) = 1 andF (k max + 1) = 0, and the function decreases monotonically along k. We can revise Code17.15 to draw CCDFs:Code 17.16: ccdfs-loglog.py