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Renormalizing complex networks

2021/05/25  |  academic

Scaling up real networks by geometric branching growth

PNAS May 25, 2021 118 (21) e2018994118; https://doi.org/10.1073/pnas.2018994118  

Branching processes underpin the complex evolution of many real systems. However, network models typically describe network growth in terms of a sequential addition of nodes. Here, we measured the evolution of real networks—journal citations and international trade—over a 100-y period and found that they grow in a self-similar way that preserves their structural features over time. This observation can be explained by a geometric branching growth model that generates a multiscale unfolding of the network by using a combination of branching growth and a hidden metric space approach. Our model enables multiple practical applications, including the detection of optimal network size for maximal response to an external influence and a finite-size scaling analysis of critical behavior.                                                                                                                                                     

Multiscale unfolding of real networks by geometric renormalization

Nature Physics doi:10.1038/s41567-018-0072-5 (2018)

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 investigation 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.