Introduction to the Modeling and Analysis of Complex Systems

402CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIESThe modularity of a given set of communities in a network is defined as follows [78]:Q = |E in| − 〈|E in |〉|E|(17.35)Here, |E| is the number of edges, |E in | is the number of within-community edges (i.e.,those that don’t cross boundaries between communities), and 〈|E in |〉 is the expectednumber of within-community edges if the topology were purely random. The subtractionof 〈|E in |〉 on the numerator penalizes trivial community structure, such as consideringthe entire network a single community that would trivially maximize |E in |.The Louvain method finds the optimal community structure that maximizes the modularityin the following steps:1. Initially, each node is assigned to its own community where the node itself is theonly community member. Therefore the number of initial communities equals thenumber of nodes.2. Each node considers each of its neighbors and evaluates whether joining to theneighbor’s community would increase the modularity of the community structure.After evaluating all the neighbors, it will join the community of the neighbor thatachieves the maximal modularity increase (only if the change is positive; otherwisethe node will remain in its own community). This will be repeatedly applied for allnodes until no more positive gain is achievable.3. The result of Step 2 is converted to a new meta-network at a higher level, by aggregatingnodes that belonged to a single community into a meta-node, representingedges that existed within each community as the weight of a self-loop attached tothe meta-node, and representing edges that existed between communities as theweights of meta-edges that connect meta-nodes.4. The above two steps are repeated until no more modularity improvement is possible.One nice thing about this method is that it is parameter-free; you don’t have to specifythe number of communities or the criteria to stop the algorithm. All you need is to providenetwork topology data, and the algorithm heuristically finds the community structure thatis close to optimal in achieving the highest modularity.Unfortunately, NetworkX doesn’t have this Louvain method as a built-in function, butits Python implementation has been developed and released freely by Thomas Aynaud,which is available from http://perso.crans.org/aynaud/communities/. Once you installit, a new community module becomes available in Python. Here is an example: