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The Stability of a Graph Partition: A Dynamics-Based Framework for Community Detection

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Dynamics On and Of Complex Networks, Volume 2

Abstract

Recent years have seen a surge of interest in the analysis of complex systems. This trend has been facilitated by the availability of relational data and the increasingly powerful computational resources that can be employed for their analysis. A unifying concept in the study of complex systems is their formalisation as networks comprising a large number of non-trivially interacting agents. By considering a network perspective, it is hoped to gain a deepened understanding of system-level properties beyond what could be achieved by focussing solely on the constituent units. Naturally, the study of real-world systems leads to highly complex networks and a current challenge is to extract intelligible, simplified descriptions from the network in terms of relevant subgraphs (or communities), which can provide insight into the structure and function of the overall system.

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Notes

  1. 1.

    In the continuous case, trace R(t) is monotonically decreasing with time. To prove this, note that \(P(t) = {D}^{-1/2}\exp (-t\mathcal{L}){D}^{1/2}\) and \(d(\text{trace }R(t))/dt = -\text{trace }{H}^{T}\Pi {D}^{-1/2}\) \({\mathcal{L}}^{1/2}\exp (-t\mathcal{L}){\mathcal{L}}^{1/2}{D}^{1/2}H = -\text{trace }{H}^{T}{D}^{1/2}{\mathcal{L}}^{1/2}\exp (-t\mathcal{L}){\mathcal{L}}^{1/2}{D}^{1/2}H/2m\). This is obviously strictly negative since the matrix exp( − ℒ) is symmetric positive definite.

  2. 2.

    In particular cases, such as a bipartite graph, trace R s can oscillate in the discrete-time case, indicating poor communities or even “anti-communities” with a rapid alternance of random walkers between communities. We therefore take the lowest point of the R s over the interval as the quality function.

  3. 3.

    Dataset taken from http://www.termoenergetica.upc.edu/marti/index.htm.

  4. 4.

    See http://de.wikipedia.org/wiki/Stromnetz%5C;#Netzbetreiber.

  5. 5.

    For a map of the French regional electrical companies see http://www.rte-france.com/fr/nous-connaitre/qui-sommes-nous/organisation-et-gouvernance/le-siege-et-les-unites-regionales.

  6. 6.

    see http://en.wikipedia.org/wiki/Autonomous_system_(Internet).

  7. 7.

    http://www.caida.org/data/active/as_taxonomy/.

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Acknowledgements

J.-C. D. acknowledges support from the grant “Actions de recherche concertées—Large Graphs and Networks” of the Communauté Française de Belgique, the EULER project (Grant No.258307) part of the Future Internet Research and Experimentation (FIRE) objective of the Seventh Framework Programme (FP7) and from the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian State Science Policy Office. S.N.Y. and M.B. acknowledge funding from grant EP/I017267/1 from the EPSRC (Engineering and Physical Sciences Research Council) of the UK under the Mathematics Underpinning the Digital Economy programme and from the Office of Naval Research (ONR) of the USA.

Author information

Authors and Affiliations

  1. Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) and Center for Operations Research and Optimisation (CORE), Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Charles Delvenne

  2. Namur Center for Complex Systems (naXys), Facultés Universitaires Notre-Dame de la Paix, Namur, Belgium

    Jean-Charles Delvenne

  3. Department of Chemistry, Imperial College London, London, UK

    Michael T. Schaub & Sophia N. Yaliraki

  4. Department of Mathematics, Imperial College London, London, UK

    Michael T. Schaub & Mauricio Barahona

Authors
  1. Jean-Charles Delvenne
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  2. Michael T. Schaub
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  3. Sophia N. Yaliraki
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  4. Mauricio Barahona
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Corresponding author

Correspondence to Jean-Charles Delvenne .

Editor information

Editors and Affiliations

  1. Dept. Computer Science & Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, West Bengal, India

    Animesh Mukherjee

  2. Microsoft Research Lab India, 9 Lavelle Road, Bangalore, 560001, India

    Monojit Choudhury

  3. , Laboratoire J.A. Dieudonné, Université de Nice - Sophie Antipolis, Parc Valrose, Nice 02, 06108, France

    Fernando Peruani

  4. Dept. Computer Science & Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, West Bengal, India

    Niloy Ganguly

  5. Department of Mathematical Engineering, Université Catholique de Louvain, 4, Avenue George Lemaître, Louvain-la-Neuve, B-1348, Belgium

    Bivas Mitra

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Delvenne, JC., Schaub, M.T., Yaliraki, S.N., Barahona, M. (2013). The Stability of a Graph Partition: A Dynamics-Based Framework for Community Detection. In: Mukherjee, A., Choudhury, M., Peruani, F., Ganguly, N., Mitra, B. (eds) Dynamics On and Of Complex Networks, Volume 2. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6729-8_11

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