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Hidden geometric correlations in real multiplex networks

Nature Physics volume 12, pages 1076–1081 (2016)Cite this article

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Abstract

Real networks often form interacting parts of larger and more complex systems. Examples can be found in different domains, ranging from the Internet to structural and functional brain networks. Here, we show that these multiplex systems are not random combinations of single network layers. Instead, they are organized in specific ways dictated by hidden geometric correlations between the layers. We find that these correlations are significant in different real multiplexes, and form a key framework for answering many important questions. Specifically, we show that these geometric correlations facilitate the definition and detection of multidimensional communities, which are sets of nodes that are simultaneously similar in multiple layers. They also enable accurate trans-layer link prediction, meaning that connections in one layer can be predicted by observing the hidden geometric space of another layer. And they allow efficient targeted navigation in the multilayer system using only local knowledge, outperforming navigation in the single layers only if the geometric correlations are sufficiently strong.

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Figure 1: Hyperbolic mapping of the IPv4/IPv6 Internet.
Figure 2: Distribution of nodes in the two-dimensional similarity space of the Internet, Drosophila and arXiv multiplexes.
Figure 3: Trans-layer connection probability in the Internet, Drosophila and arXiv multiplexes.
Figure 4: Performance of mutual navigation in synthetic multiplexes with geometric correlations.
Figure 5: Performance of mutual navigation as a function of radial and angular correlation strengths (ν, g) in a two-layer synthetic multiplex that best mimics the real IPv4/IPv6 Internet.

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Acknowledgements

This work was supported by: the European Commission through the Marie Curie ITN ‘iSocial’ grant no. PITN-GA-2012-316808; a J. S. McDonnell Foundation Scholar Award in Complex Systems; the ICREA Academia prize, funded by the Generalitat de Catalunya; the MINECO project no. FIS2013-47282-C2-1-P; and the Generalitat de Catalunya grant no. 2014SGR608. Furthermore, M.B. and M.A.S. acknowledge support from the European Commission FET-Proactive Project MULTIPLEX no. 317532.

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Authors and Affiliations

  1. Departament de Física Fonamental, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain

    Kaj-Kolja Kleineberg, Marián Boguñá & M. Ángeles Serrano

  2. Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, E-08010 Barcelona, Spain

    M. Ángeles Serrano

  3. Department of Electrical Engineering, Computer Engineering and Informatics, Cyprus University of Technology, 33 Saripolou Street, 3036 Limassol, Cyprus

    Fragkiskos Papadopoulos

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  1. Kaj-Kolja Kleineberg
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K.-K.K. and F.P. conducted the research; all authors designed the research, discussed the results, and wrote the manuscript.

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Correspondence to Kaj-Kolja Kleineberg or Fragkiskos Papadopoulos.

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Kleineberg, KK., Boguñá, M., Ángeles Serrano, M. et al. Hidden geometric correlations in real multiplex networks. Nature Phys 12, 1076–1081 (2016). https://doi.org/10.1038/nphys3812

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