Multi-scale Modularity and Dynamics in Complex Networks

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.

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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  49. [49].
    M.A. Porter, J.-P. Onnela, P.J. Mucha, Communities in networks. Not. Am. Math. Soc. 56, 1082–1097 (2009)MathSciNetMATHGoogle 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)MathSciNetMATHCrossRefGoogle 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

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