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Universality in network dynamics

Nature Physics volume 9, pages 673681 (2013) | Download Citation

  • A Corrigendum to this article was published on 05 November 2013

This article has been updated

Abstract

Despite significant advances in characterizing the structural properties of complex networks, a mathematical framework that uncovers the universal properties of the interplay between the topology and the dynamics of complex systems continues to elude us. Here we develop a self-consistent theory of dynamical perturbations in complex systems, allowing us to systematically separate the contribution of the network topology and dynamics. The formalism covers a broad range of steady-state dynamical processes and offers testable predictions regarding the system’s response to perturbations and the development of correlations. It predicts several distinct universality classes whose characteristics can be derived directly from the continuum equation governing the system’s dynamics and which are validated on several canonical network-based dynamical systems, from biochemical dynamics to epidemic spreading. Finally, we collect experimental data pertaining to social and biological systems, demonstrating that we can accurately uncover their universality class even in the absence of an appropriate continuum theory that governs the system’s dynamics.

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Change history

  • 09 October 2013

    In the version of this Article originally published, the expression for Gij on page 673 should have included an absolute value sign. In addition, the caption for Fig. 1b1–c4 was missing the final wording: (note that the bounded distributions here and throughout are normalized to have mean zero and variance one). These errors have been corrected in the HTML and PDF versions of the Article.

References

  1. 1.

    Scale-free Networks: Complex Webs in Nature and Technology (Oxford Univ. Press, 2007).

  2. 2.

    & Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford Univ. Press, 2003).

  3. 3.

    Exploring complex networks. Nature 410, 268–276 (2001).

  4. 4.

    Helbing, D., Jost, J. & Kantz, H. (eds) in Networks and Complexity (Networks and Heterogeneous Media, Vol. 3, AIMS, 2008).

  5. 5.

    Networks—An Introduction (Oxford Univ. Press, 2010).

  6. 6.

    & Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge Univ. Press, 2004).

  7. 7.

    , , & Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818 (2005).

  8. 8.

    & Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, 2010).

  9. 9.

    & Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

  10. 10.

    & Emergence of scaling in random networks. Science 286, 509–512 (1999).

  11. 11.

    & Community structure in social and biological networks. Proc. Natl Acad. Sci. USA 99, 7821–7826 (2002).

  12. 12.

    Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002).

  13. 13.

    , & Dynamical and correlation properties of the Internet. Phys. Rev. Lett. 87, 258701 (2001).

  14. 14.

    & Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).

  15. 15.

    , & Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).

  16. 16.

    , , , & Dynamic modeling of gene expression data. Proc. Natl Acad. Sci. USA 98, 1693–1698 (2001).

  17. 17.

    From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20 (2000).

  18. 18.

    , , , & Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).

  19. 19.

    & How viruses spread among computers and people. 292, 1316–1317 (2001).

  20. 20.

    , , & Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701 (2004).

  21. 21.

    , , & Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288 (2005).

  22. 22.

    & Spectral analysis and the dynamic response of complex networks. Phys. Rev. E 71, 016106 (2000).

  23. 23.

    & Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

  24. 24.

    , & Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).

  25. 25.

    & A generalized model of social and biological contagion. J. Theor. Biol. 232, 587–604 (2005).

  26. 26.

    Computational Analysis of Biochemical Systems (Cambridge Univ. Press, 2000).

  27. 27.

    & Propagation of large concentration changes in reversible protein-binding networks. Proc. Natl Acad. Sci. USA 104, 13655–13660 (2007).

  28. 28.

    & Spreading out of perturbations in reversible reaction networks. New J. Phys. 9, 273–283 (2007).

  29. 29.

    , & Fluctuations in mass-action equilibrium of protein binding networks. Phys. Rev. Lett. 101, 268102 (2008).

  30. 30.

    Handbook of Stochastic Methods (Springer, 2004).

  31. 31.

    , & Biological applications of the theory of birth-and-death processes. Brief. Bioinform. 7, 70–85 (2006).

  32. 32.

    & Modeling and Analysis of Telecommunications Networks (John Wiley, 2004).

  33. 33.

    An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall, 2006).

  34. 34.

    & Modelling and analysis of gene regulatory networks. Nature Rev. 9, 770–780 (2008).

  35. 35.

    , & General statistics of stochastic process of gene expression in eukaryotic cells. Genetics 161, 1321–1332 (2002).

  36. 36.

    , , & Making sense of microarray data distributions. Bioinformatics 18, 576–584 (2002).

  37. 37.

    , , & A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity 7, 23–40 (2003).

  38. 38.

    & Zipf’s law in gene expression. Phys. Rev. Lett. 90, 088102 (2003).

  39. 39.

    et al. Can Zipf’s law be adapted to normalize microarrays? BMC Bioinform. 6, 37–49 (2005).

  40. 40.

    , , , & Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature 427, 839–843 (2004).

  41. 41.

    , , , & Scale-free brain functional networks. Phys. Rev. Lett. 94, 018102 (2005).

  42. 42.

    , & Pacific-Asia Conference Knowledge Discovery and Data Mining (PAKDD) 380–389 (Springer, 2005).

  43. 43.

    , , , & Proc. SIAM Int. Conf. Data Mining551–556 (2007).

  44. 44.

    , & Proc. First Workshop on Online Social Networks, WOSN’08 13–18 (ACM, 2008).

  45. 45.

    , & Model for cascading failures in complex networks. Phys. Rev. E 69, 045104 (2004).

  46. 46.

    , , & Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization. Chaos 17, 026103 (2007).

  47. 47.

    The ensemble approach to understand genetic regulatory networks. Physica A 340, 733–740 (2004).

  48. 48.

    , & Entropy of dialogues creates coherent structures in e-mail traffic. Proc. Natl Acad. Sci. USA 101, 14333–14337 (2004).

  49. 49.

    et al. Identifying transcription factor functions and targets by phenotypic activation. Proc. Natl Acad. Sci. USA 103, 12045–12050 (2006).

  50. 50.

    Threshold models of collective behavior. Am. J. Soc. 83, 1420–1443 (2002).

  51. 51.

    Boolean network models of cellular regulation: Prospects and limitations. J. R. Soc. Interf. 5, S85–S94 (2008).

  52. 52.

    et al. High-quality binary protein interaction map of the yeast interactome network. Science 322, 104–110 (2008).

  53. 53.

    et al. Network motifs: Simple building blocks of complex networks. Science 298, 824–827 (2002).

  54. 54.

    Power law distributions in information science: Making the case for logarithmic binning. J. Am. Soc. Inf. Sci. Technol. 61, 2417–2425 (2010).

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Acknowledgements

We thank A. Vespignani, A. Sharma, F. Simini, J. Menche, S. Rabello, G. Ghoshal, Y-Y Liu, T. Jia, M. Pósfai, C. Song, Y-Y. Ahn, N. Blumm, D. Wang, Z. Qu, M. Schich, D. Ghiassian, S. Gil, P. Hövel, J. Gao, M. Kitsak, M. Martino, R. Sinatra, G. Tsekenis, L. Chi, B. Gabriel, Q. Jin and Y. Li for discussions, and S. S. Aleva, S. Weiss, J. de Nicolo and A. Pawling for their support. This work was supported by DARPA Grant Number 11645021; The DARPA Social Media in Strategic Communications project under agreement number W911NF-12-C-0028; the Network Science Collaborative Technology Alliance sponsored by the US Army Research Laboratory under Agreement Number W911NF-09-02-0053; the Office of Naval Research under Agreement Number N000141010968 and the Defense Threat Reduction Agency awards WMD BRBAA07-J-2-0035 and BRBAA08-Per4-C-2-0033; the National Institute of Health, Center of Excellence of Genomic Science (CEGS), Grant number NIH CEGS 1P50HG4233; and the National Institute of Health, Award number 1U01HL108630-01.

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Affiliations

  1. Center for Complex Network Research and Departments of Physics, Computer Science and Biology, Northeastern University, Boston, Massachusetts 02115, USA

    • Baruch Barzel
    •  & Albert-László Barabási
  2. Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Harvard Medical School, Boston, Massachusetts 02115, USA

    • Baruch Barzel
    •  & Albert-László Barabási
  3. Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA

    • Albert-László Barabási

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Contributions

Both authors designed and performed the research. B.B. carried out the analytical and numerical calculations and analysed the empirical data. A-L.B. was the lead writer of the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Albert-László Barabási.

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https://doi.org/10.1038/nphys2741

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