Multiscale unfolding of real networks by geometric renormalization

Symmetries in physical theories denote invariance under some transformation, such as self-similarity under a change of scale. The renormalization group provides a powerful framework to study these symmetries, leading to a better understanding of the universal properties of phase transitions. However, the small-world property of complex networks complicates application of the renormalization group by introducing correlations between coexisting scales. Here, we provide a framework for the inves- tigation of complex networks at different resolutions. The approach is based on geometric representations, which have been shown to sustain network navigability and to reveal the mechanisms that govern network structure and evolution. We define a geometric renormalization group for networks by embedding them into an underlying hidden metric space. We find that real scale-free networks show geometric scaling under this renormalization group transformation. We unfold the networks in a self- similar multilayer shell that distinguishes the coexisting scales and their interactions. This in turn offers a basis for exploring critical phenomena and universality in complex networks. It also affords us immediate practical applications, including high- fidelity smaller-scale replicas of large networks and a multiscale navigation protocol in hyperbolic space, which betters those on single layers.