Journal of Statistical Physics

, Volume 173, Issue 3–4, pp 775–782 | Cite as

Soft Communities in Similarity Space

  • Guillermo García-Pérez
  • M. Ángeles Serrano
  • Marián Boguñá


The \(\mathbb {S}^1\) model has been central in the development of the field of network geometry. It places nodes in a similarity space and connects them with a likelihood depending on an effective distance which combines similarity and popularity dimensions, with popularity directly related to the degrees of the nodes. The \(\mathbb {S}^1\) model has been mainly studied in its homogeneous regime, in which angular coordinates are independently and uniformly scattered on the circle. We now investigate if the model can generate networks with targeted topological features and soft communities, that is, inhomogeneous angular distributions. To that end, hidden degrees must depend on angular coordinates, and we propose a method to estimate them. We conclude that the model can be topologically invariant with respect to the soft-community structure. Our results expand the scope of the model beyond the independent hidden variables limit and can have an important impact in the embedding of real-world networks.


Complex networks Hidden metric spaces Similarity space Communities 



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


  1. 1.
    Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
  2. 2.
    Dorogovtsev, S.N., Mendes, J.F.F.: Accelerated Growth of Networks. Handbook of Graphs and Networks. Wiley-VCH, Berlin (2003)Google Scholar
  3. 3.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509 (1999)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Serrano, M.Á., Krioukov, D., Boguñá, M.: Self-similarity of complex networks and hidden metric spaces. Phys. Rev. Lett. 100, 078701 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Krioukov, D., Papadopoulos, F., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 80, 035101 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Boguñá, M., Papadopoulos, F., Krioukov, D.: Sustaining the internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Serrano, M.Á., Boguñá, M., Sagués, F.: Uncovering the hidden geometry behind metabolic networks. Mol. BioSyst. 8, 843 (2012)CrossRefGoogle Scholar
  8. 8.
    García-Pérez, G., Boguñá, M., Allard, A., Serrano, M.Á.: The hidden hyperbolic geometry of international trade: World Trade Atlas 1870–2013. Sci. Rep. 6, 33441 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    Radicchi, F., Castellano, C., Cecconi, F., Loreto, V., Parisi, D.: Defining and identifying communities in networks. Proc. Natl. Acad. Sci. USA 101, 2658 (2004)ADSCrossRefGoogle Scholar
  11. 11.
    Arenas, A., Fernández, A., Gómez, S.: Analysis of the structure of complex networks at different resolution levels. New J. Phys. 10(5), 053039 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Zuev, K., Boguñá, M., Bianconi, G., Krioukov, D.: Emergence of soft communities from geometric preferential attachment. Sci. Rep. 5, 9421 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Papadopoulos, F., Kitsak, M., Serrano, M.Á., Boguñá, M., Krioukov, D.: Popularity versus similarity in growing networks. Nature 489(7417), 537 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Muscoloni, A., Cannistraci, C.V.: A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities. New J. Phys. 20, 052002 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gulyás, A., Bíró, J.J., Kőrösi, A., Rétvári, G., Krioukov, D.: Navigable networks as Nash equilibria of navigation games. Nat. Commun. 6, 7651 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    García-Pérez, G., Boguñá, M., Serrano, M.A.: Multiscale unfolding of real networks by geometric renormalization. Nat. Phys. 14, 583–589 (2018)CrossRefGoogle Scholar
  18. 18.
    Allard, A., Serrano, M.Á., García-Pérez, G., Boguñá, M.: The geometric nature of weights in real complex networks. Nat. Commun. 8, 14103 (2017)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departament de Física de la Matèria CondensadaUniversitat de BarcelonaBarcelonaSpain
  2. 2.Universitat de Barcelona Institute of Complex Systems (UBICS)Universitat de BarcelonaBarcelonaSpain
  3. 3.ICREABarcelonaSpain

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