Optimal map of the modular structure of complex networks

The 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 the 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 the 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 much data as the number of modules times the 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 then we use truncated singular value decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allows us to scrutinize the skeleton of the modular structure, revealing the structure of individual modules and their interrelations.