Advertisement

Multi-scale Modularity and Dynamics in Complex Networks

  • Renaud Lambiotte
Chapter
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.

Notes

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).

References

  1. [1].
    D.M. Abrams, R. Mirollo, S.H. Strogatz, D.A. Wiley, Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101, 084103 (2008)CrossRefGoogle Scholar
  2. [2].
    Y.Y. Ahn, J.P. Bagrow, S. Lehmann, Communities and hierarchical organization of links in complex networks. Nature 466, 761 (2010)CrossRefGoogle Scholar
  3. [3].
    A. Arenas, A. Díaz-Guilera, C.J. Pérez-Vicente, Synchronization reveals topological scales in complex networks. Phys. Rev. Lett. 96, 114102 (2006)CrossRefGoogle Scholar
  4. [4].
    A. Arenas, A. Fernández, S. Gómez, Analysis of the structure of complex networks at different resolution levels. New J. Phys. 10, 053039 (2008)CrossRefGoogle Scholar
  5. [5].
    M.J. Barber, Modularity and community detection in bipartite networks. Phys. Rev. E 76, 066102 (2007)MathSciNetCrossRefGoogle Scholar
  6. [6].
    M. Batty, K.J. Tinkler, Symmetric structure in spatial and social processes. Env. Plan. B 6, 3 (1979)CrossRefGoogle Scholar
  7. [7].
    V.D. Blondel, J.-L. Guillaume, R. Lambiotte, E. Lefebvre, Fast unfolding of communities in large networks. J. Stat. Mech. P10008 (2008)Google Scholar
  8. [8].
    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  9. [9].
    U. Brandes, D. Delling, M. Gaertler, R. Goerke, M. Hoefer, Z. Nikoloski, D. Wagner, Maximizing modularity is hard. arXiv:physics/0608255Google Scholar
  10. [10].
    A. Clauset, C. Moore, M.E.J. Newman, Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008)CrossRefGoogle Scholar
  11. [11].
    L. Danon, J. Duch, A. Diaz-Guilera, A. Arenas, Comparing community structure identification. J. Stat. Mech. P09008 (2005)Google Scholar
  12. [12].
    A. Delmotte, E.W. Tate, S.N. Yaliraki, M. Barahona, Protein multi-scale organization through graph partitioning and robustness analysis: application to the myosin-myosin light chain interaction. Phys. Biol. 8, 055010 (2011)CrossRefGoogle Scholar
  13. [13].
    J.-C. Delvenne, S. Yaliraki, M. Barahona, Stability of graph communities across time scales. Proc. Natl. Acad. Sci. USA 107, 12755–12760 (2010)CrossRefGoogle Scholar
  14. [14].
    J.-C. Delvenne, M.T. Schaub, S.N. Yaliraki, M. Barahona, The stability of a graph partition: a dynamics-based framework for community detection, in Dynamics on and of Complex Networks, vol. 2: Applications to Time-Varying Dynamical Systems, ed. by A. Mukherjee, M. Choudhury, F. Peruani, N. Ganguly, B. Mitra (Springer, New York, 2013)Google Scholar
  15. [15].
    T.S. Evans, Complex networks. Contemp. Phys. 45, 455 (2004)CrossRefGoogle Scholar
  16. [16].
    T.S. Evans, R. Lambiotte, Line graphs, link partitions and overlapping communities. Phys. Rev. E 80, 016105 (2009)CrossRefGoogle Scholar
  17. [17].
    D.J. Fenn, M.A. Porter, M. McDonald, S. Williams, N.F. Johnson, N.S. Jones, Dynamic communities in multichannel data: an application to the foreign exchange market during the 2007–2008 credit crisis. Chaos 19, 033119 (2009)CrossRefGoogle Scholar
  18. [18].
    M. Fiedler, A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech. Math. J. 25, 619–633 (1975)MathSciNetGoogle Scholar
  19. [19].
    S. Fortunato, M. Barthélemy, Resolution limit in community detection. Proc. Natl. Acad. Sci. USA 104, 36 (2007)CrossRefGoogle Scholar
  20. [20].
    S. Fortunato, Community detection in graphs. Phys. Rep. 486, 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  21. [21].
    B.H. Good, Y.-A. de Montjoye, A. Clauset, The performance of modularity maximization in practical contexts. Phys. Rev. E 81, 046106 (2010)MathSciNetCrossRefGoogle Scholar
  22. [22].
    T. Gross, B. Blasius, Adaptive coevolutionary networks: a review. J. R. Soc. Interface 5, 259–271 (2008)CrossRefGoogle Scholar
  23. [23].
    R. Guimerá, M. Sales, L.A.N. Amaral, Modularity from fluctuations in random graphs and complex networks. Phys. Rev. E 70, 025101 (2004)CrossRefGoogle Scholar
  24. [24].
    B. Karrer, E. Levina, M.E.J. Newman, Robustness of community structure in networks. Phys. Rev. E 77, 046119 (2008)CrossRefGoogle Scholar
  25. [25].
    N. Kashtan, U. Alon, Spontaneous evolution of modularity and network motifs. Proc. Natl. Acad. Sci. USA 102, 13773–13778 (2005)CrossRefGoogle Scholar
  26. [26].
    J.M. Kumpula, J. Saramäki, K. Kaski, J. Kertész, Limited resolution in complex network community detection with Potts model approach. Eur. Phys. J. B 56, 41–45 (2007)CrossRefGoogle Scholar
  27. [27].
    R. Lambiotte, M. Ausloos, J.A. Holyst, Majority model on a network with communities. Phys. Rev. E 75, 030101 (2007)CrossRefGoogle Scholar
  28. [28].
    R. Lambiotte, J.-C. Delvenne, M. Barahona, Laplacian dynamics and multiscale modular structure in networks. arXiv:0812.1770Google Scholar
  29. [29].
    R. Lambiotte, Multi-scale modularity in complex networks, in Proceedings of the 8th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), pp. 546–553, 2010Google Scholar
  30. [30].
    R. Lambiotte, P. Panzarasa, Communities, knowledge creation and information diffusion. J. Informetrics 3, 180–190 (2009)CrossRefGoogle Scholar
  31. [31].
    R. Lambiotte, J.C. Gonzalez-Avella, On co-evolution and the importance of initial conditions. Phys. A 390, 392–397 (2011)CrossRefGoogle Scholar
  32. [32].
    R. Lambiotte, R. Sinatra, J.-C. Delvenne, T.S. Evans, M. Barahona, V. Latora, Flow graphs: interweaving dynamics and structure. Phys. Rev. E 84 017102 (2011)CrossRefGoogle Scholar
  33. [33].
    R. Lambiotte, M. Rosvall, Ranking and clustering of nodes in networks with smart teleportation. Phys. Rev. E 85, 056107 (2012)CrossRefGoogle Scholar
  34. [34].
    A. Lancichinetti, S. Fortunato, J. Kertész, Detecting the overlapping and hierarchical community structure of complex networks. New J. Phys. 11, 033015 (2009)CrossRefGoogle Scholar
  35. [35].
    A. Lancichinetti, S. Fortunato, Community detection algorithms: a comparative analysis. Phys. Rev. E 80, 056117 (2009)CrossRefGoogle Scholar
  36. [36].
    A. Lancichinetti, F. Radicchi, J.J. Ramasco, S. Fortunato, Finding statistically significant communities in networks. PLoS ONE 6, e18961 (2011)CrossRefGoogle Scholar
  37. [37].
    D. Meunier, R. Lambiotte, A. Fornito, K.D. Ersche, E.T. Bullmore, Hierarchical modularity in human brain functional networks. Front. Neuroinform. 3, 37 (2009)CrossRefGoogle Scholar
  38. [38].
    D. Meunier, R. Lambiotte, E.T. Bullmore, Modular and hierarchically modular organization of brain networks. Front. Neurosci. 4, 200 (2010)CrossRefGoogle Scholar
  39. [39].
    M.E.J. Newman, The structure and function of complex networks. SIAM Rev. 45, 167 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40].
    M.E.J. Newman, M. Girvan, Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004)CrossRefGoogle Scholar
  41. [41].
    M.E.J. Newman, Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103, 8577 (2006)CrossRefGoogle Scholar
  42. [42].
    M.E.J. Newman, Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74, 036104 (2006)MathSciNetCrossRefGoogle Scholar
  43. [43].
    V. Nicosia, G. Mangioni, V. Carchiolo, M. Malgeri, Extending the definition of modularity to directed graphs with overlapping communities. J. Stat. Mech. P03024 (2009)Google Scholar
  44. [44].
    G. Palla, I. Derenyi, I. Farkas, T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818 (2005)CrossRefGoogle Scholar
  45. [45].
    R.K. Pan, S. Sinha, Modular networks with hierarchical organization: The dynamical implications of complex structure. Pramana J. Phys. 71, 331–340 (2008)CrossRefGoogle Scholar
  46. [46].
    R.K. Pan, S. Sinha, Modularity produces small-world networks with dynamical time-scale separation. Europhys. Lett. 85, 68006 (2009)CrossRefGoogle Scholar
  47. [47].
    R.K. Pan, N. Chatterjee, S. Sinha, Mesoscopic organization reveals the constraints governing Caenorhabditis elegans nervous system. PLoS ONE 5, e9240 (2010)CrossRefGoogle Scholar
  48. [48].
    P. Pons, M. Latapy, Computing communities in large networks using random walks. J. Graph Algorithm Appl. 10, 191 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49].
    M.A. Porter, J.-P. Onnela, P.J. Mucha, Communities in networks. Not. Am. Math. Soc. 56, 1082–1097 (2009)MathSciNetzbMATHGoogle Scholar
  50. [50].
    J. Reichardt, S. Bornholdt, Statistical mechanics of community detection. Phys. Rev. E 74, 016110 (2006)MathSciNetCrossRefGoogle Scholar
  51. [51].
    P.A. Robinson, J.A. Henderson, E. Matar, P. Riley, R.T. Gray, Dynamical reconnection and stability constraints on cortical network architecture. Phys. Rev. Lett. 103, 108104 (2009)CrossRefGoogle Scholar
  52. [52].
    P. Ronhovde, Z. Nussinov, Multiresolution community detection for megascale networks. Phys. Rev. E 80, 016109 (2009)CrossRefGoogle Scholar
  53. [53].
    M. Rosvall, C.T. Bergstrom, Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. USA 105, 1118 (2008)CrossRefGoogle Scholar
  54. [54].
    M. Rosvall, D. Axelsson, C.T. Bergstrom, The map equation. Eur. Phys. J. Spec. Top. 178, 13–23 (2009)CrossRefGoogle Scholar
  55. [55].
    M. Rosvall, C.T. Bergstrom, Multilevel compression of random walks on networks reveals hierarchical organization in large integrated systems. PLoS ONE 6, e18209 (2011)CrossRefGoogle Scholar
  56. [56].
    M. Sales-Pardo, R. Guimerá, A. Moreira, L.A.N. Amaral, Extracting the hierarchical organization of complex systems. Proc. Natl. Acad. Sci. USA 104, 15224 (2007)CrossRefGoogle Scholar
  57. [57].
    M.T. Schaub, J.-C. Delvenne, S.N. Yaliraki, M. Barahona, Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit. PLoS ONE 7, e32210 (2012)CrossRefGoogle Scholar
  58. [58].
    M.T. Schaub, R. Lambiotte, M. Barahona, Encoding dynamics for multiscale community detection: Markov time sweeping for the map equation. Phys. Rev. E 86, 026112. Published 21 August 2012Google Scholar
  59. [59].
    M. Shanahan, Dynamical complexity in small-world networks of spiking neurons. Phys. Rev. E 78, 041924 (2008)MathSciNetCrossRefGoogle Scholar
  60. [60].
    M. Shanahan, Metastable chimera states in community-structured oscillator networks. Chaos 20, 013108 (2010)MathSciNetCrossRefGoogle Scholar
  61. [61].
    J. Shi, J. Malik, Normalized cuts and image segmentation. IEEE Trans. Patt. Anal. Mach. Intell. 22, 888 (2000)CrossRefGoogle Scholar
  62. [62].
    H.A. Simon, The architecture of complexity. Proc. Am. Phil. Soc. 106, 467–482 (1962)Google Scholar
  63. [63].
    H.A. Simon, Near-decomposability and complexity: How a mind resides in a brain, in The Mind, the Brain, and Complex Adaptive Systems, ed. by H. Morowitz, J. Singer (Addison-Wesley, Reading, 1995), pp. 25–43Google Scholar
  64. [64].
    O. Sporns, G. Tononi, G. Edelman, Theoretical neuroanatomy: Relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb Cortex 10, 127–141 (2000)CrossRefGoogle Scholar
  65. [65].
    O. Sporns, D. Chialvo, M. Kaiser, C.C. Hilgetag, Organization, development and function of complex rain networks. Trends Cognit. Sci. 8, 418–425 (2004)CrossRefGoogle Scholar
  66. [66].
    V.A. Traag, P. Van Dooren, Y. Nesterov, Narrow scope for resolution-limit-free community detection. Phys. Rev. E 84, 016114 (2011)CrossRefGoogle Scholar
  67. [67].
    J.N. Tsitsiklis, Problems in decentralized decision making and computation. Ph.D. thesis, MIT (1984)Google Scholar
  68. [68].
    S. Van Dongen, Graph clustering via a discrete uncoupling process. SIAM. J. Matrix Anal. Appl. 30, 121–141 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  69. [69].
    W.X. Wang, B.-H. Wang, C.-Y. Yin, Y.-B. Xie, T. Zhou, Traffic dynamics based on local routing protocol on a scale-free network. Phys. Rev. E 73, 026111 (2006)CrossRefGoogle Scholar
  70. [70].
    D.J. Watts, S.H. Strogatz, Collective dynamics of small-world networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of NamurNamurBelgium

Personalised recommendations