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The Power of Consensus: Random Graphs Still Have No Communities

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Social Network Analysis - Community Detection and Evolution

Part of the book series: Lecture Notes in Social Networks ((LNSN))

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Abstract

Communities are a powerful tool to describe the structure of complex networks. Algorithms aiming at maximizing a quality function called modularity have been shown to effectively compute the community structure. However, some problems remain: in particular, it is possible to find high modularity partitions in graph without any community structure, in particular random graphs. In this paper, we study the notion of consensual communities, or community cores, and show that they do not exist in random graphs. For that, we exhibit a phase transition based on the strength of consensus: below a given threshold, all the nodes belongs to the same consensual community; above this threshold, each node is in its own consensual community. We compare the results using different quality functions as well as different models of random graphs, with or without communities.

The work presented in this paper is an extension of [6].

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Notes

  1. 1.

    Also, an execution takes less than one hour on a network with more than one billion of nodes and links.

  2. 2.

    Assumptions in classical mean field make extensive use of the fact that a random graph whose size tends to infinity is locally a tree.

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Acknowledgments

We would like to thank the anonymous referees for their insightful comments and suggestions, which have helped to improve the presentation of this paper. This work is partially supported by the DynGraph ANR-10-JCJC-0202 and CODDDE ANR-13-CORD-0017-01 projects of the French National Research Agency.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6, 75005, Paris, France

    Romain Campigotto & Jean-Loup Guillaume

  2. CNRS, UMR 7606, LIP6, 75005, Paris, France

    Romain Campigotto & Jean-Loup Guillaume

Authors
  1. Romain Campigotto
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  2. Jean-Loup Guillaume
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Corresponding author

Correspondence to Jean-Loup Guillaume .

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Editors and Affiliations

  1. Département d'Informatique et Ingénirie, Université du Québec en Outaouais, Gatineau, Québec, Canada

    Rokia Missaoui

  2. Département de Mathématiques et Informatique, Université Cheikh Anta Diop, Dakar, Senegal

    Idrissa Sarr

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Campigotto, R., Guillaume, JL. (2014). The Power of Consensus: Random Graphs Still Have No Communities. In: Missaoui, R., Sarr, I. (eds) Social Network Analysis - Community Detection and Evolution. Lecture Notes in Social Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-12188-8_7

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  • DOI: https://doi.org/10.1007/978-3-319-12188-8_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12187-1

  • Online ISBN: 978-3-319-12188-8

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