Laplacian Coarse Graining in Complex Networks

Abstract: Complex networks can model a range of different systems, from the human brain to social connections. Some of those networks have a large number of nodes and links, making it impractical to analyze them directly. One strategy to simplify these systems is by creating miniaturized versions of the networks that keep their main properties. A convenient tool that applies that strategy is the renormalization group (RG), a methodology used in statistical physics to change the scales of physical systems. This method consists of two steps: a coarse grain, where one reduces the size of the system, and a rescaling of the interactions to compensate for the information loss. This work applies RG to complex networks by introducing a coarse-graining method based on the Laplacian matrix. We use a field-theoretical approach to calculate the correlation function and coarse-grain the most correlated nodes into super-nodes, applying our method to several artificial and real-world networks. The results are promising, with most of the networks under analysis showing self-similar properties across different scales.