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Multi-scale Modularity and Dynamics in Complex Networks

Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

A broad range of systems are made of elements in interaction and can be represented as networks. Important examples include social networks, the Internet, airline routes, and a wide range of biological networks.

Keywords

  • Random Walk
  • Null Model
  • Adjacency Matrix
  • Quality Function
  • Community Detection

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Strictly speaking, the normalized Laplacian of a network is \(L_{ij}^{^{\prime}} = A_{ij}/(k_{i}^{1/2}k_{j}^{1/2}) -\delta _{ij}\), but L and L are equivalent by similarity as \(L_{ij}^{^{\prime}} = k_{i}^{-1/2}L_{ij}k_{j}^{1/2}\).

  2. 2.

    In a nutshell, this size dependence originates from a choice of null model where each pair of nodes i and j can be connected, given a certain number of available links m in the system, whatever the distance between i and j in the network. Local null models where pairs of nodes are randomly connected only within a finite radius of interaction are expected to hinder this effect.

  3. 3.

    Ergodicity is ensured if the underlying network is non-bipartite and connected.

  4. 4.

    Modularity was first proposed to find the best partition in a nested hierarchy of possible community divisions [40].

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Acknowledgements

I would like to thank my coauthors involved in the articles described in this chapter, in particular J.-C. Delvenne and M. Barahona for [28], M.T. Schaub for [58], and D. Meunier and E.T Bullmore for [38]. Some of the ideas presented in this chapter are developed in another chapter of this book [14]. I would also like to acknowledge support from FNRS (MIS-2012-F.4527.12) and Belspo (PAI Dysco).

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Lambiotte, R. (2013). Multi-scale Modularity and Dynamics in Complex Networks. 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_7

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