Geometric renormalization of complex networks

Complex networks are small-world, strongly clustered and hierarchical, modular, robust yet fragile, and may exhibit unexpected responses like cascades and other critical and extreme events. Many of these fundamental properties are well explained by a family of hidden metric space network models that led to the discovery that the latent geometry of many real networks is hyperbolic. Hyperbolicity emerges as a result of the combination of popularity and similarity dimensions into an effective distance between nodes, such that more popular and similar nodes have more chance to interact. The geometric approach permits the production of truly cartographic maps of real networks that are not only visually appealing, but enable applications like efficient navigation and the detection of communities of similar nodes. Recently, it has also enabled the introduction of a geometric renormalization group that unravels the multiple length scales coexisting in complex networks, strongly intertwined due to their small world property. Interestingly, many real-world networks are self-similar when observed at the different resolutions unfolded by geometric renormalization, a property that may find its origin in a common growth mechanism. Practical applications of the geometric renormalization group for networks include high-fidelity downscaled or upscaled network replicas, and many others.