Network renormalization

The renormalization group (RG) is a powerful theoretical framework. It is used on systems with many degrees of freedom to transform the description of their configurations, along with the associated model parameters and coupling constants, across different levels of resolution. The RG also provides a way to identify critical points of phase transitions and study the system’s behaviour around them. In traditional physical applications, the RG largely builds on the notions of homogeneity, symmetry, geometry and locality to define metric distances, scale transformations and self-similar coarse-graining schemes. More recently, efforts have been made to extend RG concepts to complex networks. However, in such systems, explicit geometric coordinates do not necessarily exist, different nodes and subgraphs can have different statistical properties, and homogeneous lattice-like symmetries are absent — all features that make it complicated to define consistent renormalization procedures. In this Technical Review, we discuss the main approaches, important advances, and the remaining open challenges for network renormalization.