Network Renormalizaton

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 coarse-graining 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.