Introduction to the Modeling and Analysis of Complex Systems

400CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESand biological networks may be explained by this simple structural reason. In order todetermine whether or not a network showing negative assortativity is fundamentally disassortativefor non-structural reasons, you will need to conduct a control experiment byrandomizing its topology and measuring assortativity while keeping the same degree distribution.Exercise 17.14 Randomize the topology of the Karate Club graph while keepingits degree sequence, and then measure the degree assortativity coefficient ofthe randomized graph. Repeat this many times to obtain a distribution of the coefficientsfor randomized graphs. Then compare the distribution with the actualassortativity of the original Karate Club graph. Based on the result, determinewhether or not the Karate Club graph is truly assortative or disassortative.17.7 Community Structure and ModularityThe final topics of this chapter are the community structure and modularity of a network.These topics have been studied very actively in network science for the last several years.These are typical mesoscopic properties of a network; neither microscopic (e.g., degreesor clustering coefficients) nor macroscopic (e.g., density, characteristic path length) propertiescan tell us how a network is organized at spatial scales intermediate between thosetwo extremes, and therefore, these concepts are highly relevant to the modeling and understandingof complex systems too.Community A set of nodes that are connected more densely to each other than tothe rest of the network. Communities may or may not overlap with each other,depending on their definitions.Modularity The extent to which a network is organized into multiple communities.Figure 17.13 shows an example of communities in a network.There are literally dozens of different ways to define and detect communities in anetwork. But here, we will discuss just one method that is now widely used by networkscience researchers: the Louvain method, proposed by Vincent Blondel et al. in 2008[77]. It is a very fast, efficient heuristic algorithm that maximizes the modularity of nonoverlappingcommunity structure through an iterative, hierarchical optimization process.