Size reduction of complex networks preserving modularity

Abstract

The ubiquity of modular structure in real-world complex networks is being the focus of attention in many trials to understand the interplay between network topology and functionality. The best approaches to the identification of modular structure are based on the optimization of a quality function known as modularity. However this optimization is a hard task provided that the computational complexity of the problem is in the NP-hard class. Here we propose an exact method for reducing the size of weighted (directed and undirected) complex networks while maintaining invariant its modularity. This size reduction allows the heuristic algorithms that optimize modularity for a better exploration of the modularity landscape. We compare the modularity obtained in several real complex-networks by using the Extremal Optimization algorithm, before and after the size reduction, showing the improvement obtained. We speculate that the proposed analytical size reduction could be extended to an exact coarse graining of the network in the scope of real-space renormalization.

Fast unfolding of communities in large networks

“…between the new nodes are given by the sum of the weight of the links between nodes in the corresponding two communities [21]. Links between nodes of the same community lead to self-loops for this community in the new network.…”

[Paper] An Efficient Extraction Method of Hierarchical Structure of Web Communities for Web Video Retrieval

“…First, we construct a "community graph" of which each node consists of multiple Web videos, and each edge corresponds to hyperlinks between Web pages including these videos by using the Web video features ζ . Then hierarchical structure of Web communities containing Web videos that have similar topics is estimated on the basis of SCCs 14) , edge betweenness 15) and modularity 16) of the community graph. The above procedures are explained in detail below.…”

“…It should be noted that when all edges in G are cut, each child Web community becomes equivalent to each Web video set, i.e., each node of the community graph G. Since it is not effective to provide all Web communities until all edges in G have been cut when Web video retrieval is performed, we have to decide how many hierarchies of the Web communities to provide. In order to meet this requirement, our method uses modularity 16) , i.e., a quality function that evaluates the division results of communities in a graph. Suppose that Q Ncut is modularity of G, where the number of cut edges is N cut , and Q Ncut is defined as follows 16) :…”

“…In order to meet this requirement, our method uses modularity 16) , i.e., a quality function that evaluates the division results of communities in a graph. Suppose that Q Ncut is modularity of G, where the number of cut edges is N cut , and Q Ncut is defined as follows 16) :…”

“…Furthermore, we define a community graph of which each node consists of multiple Web videos and each edge corresponds to hyperlinks between Web pages including these videos. Then hierarchical structure of Web communities is estimated on the basis of SCCs 14) , edge betweenness 15) and modularity 16) of the community graph. Since many Web videos can be handled by the community graph with a small number of nodes, the number of computation targets for edge betweenness is reduced.…”