Introduction to the Modeling and Analysis of Complex Systems

17.6. ASSORTATIVITY 397Assortativity (positive assortativity) The tendency for nodes to connect to othernodes with similar properties within a network.Disassortativity (negative assortativity) The tendency for nodes to connect to othernodes with dissimilar properties within a network.Assortativity coefficient∑(i,j)∈Er =(f(i) − ¯f 1 )(f(j) − ¯f 2 )√ ∑(i,j)∈E (f(i) − ¯f√ ∑1 ) 2 (i,j)∈E (f(j) − ¯f(17.33)2 ) 2where E is the set of directed edges (undirected edges should appear twice inE in two directions), and∑(i,j)∈E¯f 1 =f(i) ∑(i,j)∈E, ¯f2 =f(j). (17.34)|E||E|The assortativity coefficient is a Pearson correlation coefficient of some nodeproperty f between pairs of connected nodes. Positive coefficients imply assortativity,while negative ones imply disassortativity.If the measured property is a node degree (i.e., f = deg), this is called thedegree assortativity coefficient. For directed networks, each of ¯f 1 and ¯f 2 can beeither in-degree or out-degree, so there are four different degree assortativitiesyou can measure: in-in, in-out, out-in, and out-out.Let’s look at some example. Here is how to draw a degree-degree scatter plot:Code 17.19: degree-correlation.pyfrom pylab import *import networkx as nxn = 1000ba = nx.barabasi_albert_graph(n, 5)xdata = []ydata = []for i, j in ba.edges_iter():xdata.append(ba.degree(i)); ydata.append(ba.degree(j))