Optimal map of the modular structure of complex networks

TL;DR

This paper introduces a mathematical approach using contribution matrices and truncated singular value decomposition to analyze and visualize the modular structure of complex networks, revealing interrelations among modules.

Contribution

It presents a novel method to dissect and visualize the modular structure of complex networks through contribution matrices and SVD, providing clearer insights into module interrelations.

Findings

Effective visualization of module interrelations

Identification of individual module structures

Enhanced understanding of network modularity

Abstract

Modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and function of complex systems. Generally speaking, modules are islands of highly connected nodes separated by a relatively small number of links. Every module can have contributions of links from any node in the network. The challenge is to disentangle these contributions to understand how the modular structure is built. The main problem is that the analysis of a certain partition into modules involves, in principle, as many data as number of modules times number of nodes. To confront this challenge, here we first define the contribution matrix, the mathematical object containing all the information about the partition of interest, and after, we use a Truncated Singular Value Decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allow us to scrutinize the skeleton of the modular structure, revealing the structure of individual modules and their interrelations.