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Multiscale unfolding of real networks by geometric renormalization

Nature Physicsvolume 14pages583589 (2018) | Download Citation

Abstract

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.

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Acknowledgements

We acknowledge support from a James S. McDonnell Foundation Scholar Award in Complex Systems, the ICREA Academia prize, funded by the Generalitat de Catalunya, and Ministerio de Economa y Competitividad of Spain projects no. FIS2013-47282-C2-1-P and no. FIS2016-76830-C2-2-P (AEI/FEDER, UE).

Author information

Affiliations

  1. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain

    • Guillermo García-Pérez
    • , Marián Boguñá
    •  & M. Ángeles Serrano
  2. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain

    • Guillermo García-Pérez
    • , Marián Boguñá
    •  & M. Ángeles Serrano
  3. ICREA, Barcelona, Spain

    • M. Ángeles Serrano

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Contributions

G.G.-P., M.B. and M.Á.S. contributed to the design and implementation of the research, the analysis of the results and the writing of the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to M. Ángeles Serrano.

Supplementary information

  1. Supplementary Information

    Supplementary notes, supplementary figures 1–16, supplementary references

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DOI

https://doi.org/10.1038/s41567-018-0072-5