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Systems with lattice geometry can be renormalized exploiting their embedding in metric space, which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual techniques because of their small-world character and lack of explicit metric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, thereby allowing for the definition of `block nodes across multiple scales. This approach reveals a necessary and specific dependence of network topology on an additive hidden variable attached to nodes, plus optional dyadic factors. Renormalizable networks turn out to be consistent with a unique specification of the fitness model, while they are incompatible with preferential attachment, the configuration model or the stochastic blockmodel. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization. If the hidden variables are annealed, the model spontaneously leads to realistic scale-free networks with cut-off. If they are quenched, the model can be used to renormalize real-world networks with node attributes and distance-dependence or communities. As an example we derive an accurate multiscale model of the International Trade Network applicable across hierarchical geographic partitions.
Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effectiv
A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years,
A valid prediction for a physical observable from quantum field theory should be independent of the choice of renormalization scheme -- this is the primary requirement of renormalization group invariance (RGI). Satisfying scheme invariance is a chall
Clustering $unicode{x2013}$ the tendency for neighbors of nodes to be connected $unicode{x2013}$ quantifies the coupling of a complex network to its underlying latent metric space. In random geometric graphs, clustering undergoes a continuous phase t
To understand the formation, evolution, and function of complex systems, it is crucial to understand the internal organization of their interaction networks. Partly due to the impossibility of visualizing large complex networks, resolving network str