The renormalization group (RG) is a powerful theoretical framework for studying the multiscale nature of systems with many interacting components, in which correlations span a wide range of scales. It works by systematically “zooming out”, transforming how we describe a system across different levels of resolution. RG is also central to identifying critical points of phase transitions and to understanding how a system behaves near criticality.
In traditional physics setting, RG relies heavily on properties such as homogeneity, symmetry, and locality, which make it possible to define metric distances, scale transformations, and self-similar coarsegraining procedures in a consistent way. Extending these RG ideas to complex networks is challenging: networks often lack explicit geometric embedding; different nodes and subgraphs can have different statistical properties; and the regular, spatial or lattice-like symmetries that RG usually exploits are typically absent. In this Technical Review, we survey the main approaches to network renormalization, highlight the geometric renormalization group as a key advance, and outline the most important open challenges ahead.
