Introduction to the Modeling and Analysis of Complex Systems

372CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESThe density of a network is the fraction between 0 and 1 that tells us what portion ofall possible edges are actually realized in the network. For a network G made of nnodes and m edges, the density ρ(G) is given byρ(G) =mn(n−1)2=2mn(n − 1)(17.1)for an undirected network, orρ(G) =mn(n − 1)(17.2)for a directed network.NetworkX has a built-in function to calculate network density:Code 17.2:>>> g = nx.karate_club_graph()>>> nx.density(g)0.13903743315508021Note that the size and density of a network don’t specify much about the network’s actualtopology (i.e., shape). There are many networks with different topologies that have exactlythe same size and density.But there are some things the size and density can still predict about networks. Onesuch example is network percolation, i.e., whether or not the nodes are sufficiently connectedto each other so that they form a giant component that is visible at macroscopicscales. I can show you an example. In the code below, we generate Erdős-Rényi randomgraphs made of 100 nodes with different connection probabilities:Code 17.3: net-percolation.pyfrom pylab import *import networkx as nxfor i, p in [(1, 0.0001), (2, 0.001), (3, 0.01), (4, 0.1)]:subplot(1, 4, i)title(’p = ’ + str(p))g = nx.erdos_renyi_graph(100, p)nx.draw(g, node_size = 10)