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Soft Communities in Similarity Space

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Abstract

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.

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Notes

  1. The original model defined in [4] is in fact more general, allowing for any connection probability \(p_{ij}\) as long as it depends on the argument \(d_{ij}/(\kappa _i \kappa _j)^{1/D}\), where the space is the D-dimensional sphere and \(d_{ij}\) the geodesic distance on the sphere. The particular functional form in Eq. (1) allows us to interpret the network as a set of non-interacting fermions (the links) embedded in the hyperbolic plane, with the hyperbolic length of a link playing the role of its energy and \(\beta \) playing the role of the inverse of the bath temperature [5].

  2. The ordering of the target degrees might not be necessary in a more general situation where, for instance, hidden degrees are not correlated with angles.

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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; Ministerio de Economía y Competitividad of Spain Projects No. FIS2013-47282-C2-1-P and no. FIS2016-76830-C2-2-P (AEI/FEDER, UE).

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Authors and Affiliations

  1. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028, Barcelona, Spain

    Guillermo García-Pérez, M. Ángeles Serrano & Marián Boguñá

  2. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain

    Guillermo García-Pérez, M. Ángeles Serrano & Marián Boguñá

  3. ICREA, Pg. Lluís Companys 23, 08010, Barcelona, Spain

    M. Ángeles Serrano

Authors
  1. Guillermo García-Pérez
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  2. M. Ángeles Serrano
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  3. Marián Boguñá
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Correspondence to M. Ángeles Serrano.

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García-Pérez, G., Serrano, M.Á. & Boguñá, M. Soft Communities in Similarity Space. J Stat Phys 173, 775–782 (2018). https://doi.org/10.1007/s10955-018-2084-z

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