Spatiotemporal signal propagation in complex networks

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A major achievement in the study of complex networks is the realization that diverse systems, from sub-cellular biology to social networks, exhibit universal topological characteristics. Yet, such universality does not naturally translate to the dynamics of these systems, as dynamic behaviour cannot be uniquely predicted from topology alone. Rather, it depends on the interplay of the network’s topology with the dynamic mechanisms of interaction between the nodes. Hence, systems with similar structure may exhibit profoundly different dynamic behaviour. We therefore seek a general theoretical framework to help us systematically translate topological elements into their predicted dynamic outcome. Here, we offer such a translation in the context of signal propagation, linking the topology of a network to the observed spatiotemporal spread of perturbative signals across it, thus capturing the network’s role in propagating local information. For a range of nonlinear dynamic models, we predict that the propagation rules condense into three highly distinctive dynamic regimes, characterized by the interplay between network paths, degree distribution and the interaction dynamics. As a result, classifying a system’s intrinsic interaction mechanisms into the relevant dynamic regime allows us to systematically translate topology into dynamic patterns of information propagation.

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Fig. 1: Propagation of signals in a complex network environment.
Fig. 2: Testing ground and characterization of network signal propagation.
Fig. 3: Classifying the zoo of propagation patterns.
Fig. 4: Dynamic regimes of signal propagation.
Fig. 5: Propagation between communities (see Methods).
Fig. 6: The universal temporal distance \({\cal L}(j \to i)\).

Data availability

All data and codes to reproduce the results presented here are freely accessible at Additional information is available from the corresponding author upon reasonable request. The only exception is the air-traffic network data, which the authors are restricted from sharing.


  1. 1.

    Brockmann, D. & Helbing, D. The hidden geometry of complex, network-driven contagion phenomena. Science 342, 1337–1342 (2013).

  2. 2.

    Kumar, J., Rotter, S. & Aertsen, A. Spiking activity propagation in neuronal networks: reconciling different perspectives on neural coding. Nat. Rev. Neurosci. 11, 615–627 (2010).

  3. 3.

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

  4. 4.

    Barzel, B. & Barabási, A.-L. Universality in network dynamics. Nat. Phys. 9, 673–681 (2013).

  5. 5.

    Barzel, B., Liu, Y.-Y. & Barabási, A.-L. Constructing minimal models for complex system dynamics. Nat. Commun. 6, 7186 (2015).

  6. 6.

    Holter, N. S., Maritan, A., Cieplak, M., Fedoroff, N. V. & Banavar, J. R. Dynamic modeling of gene expression data. Proc. Natl Acad. Sci. USA 98, 1693–1698 (2001).

  7. 7.

    Afraimovich, V. S. & Bunimovich, L. A. Dynamical networks: interplay of topology, interactions and local dynamics. Nonlinearity 20, 1761–1771 (2007).

  8. 8.

    Kirst, C., Timme, M. & Battaglia, D. Dynamic information routing in complex networks. Nat. Commun. 7, 11061 (2016).

  9. 9.

    Barrat, A., Barthélemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, Cambridge, 2008).

  10. 10.

    Gao, J., Barzel, B. & Barabási, A.-L. Universal resilience patterns in complex networks. Nature 530, 307–312 (2016).

  11. 11.

    Harush, U. & Barzel, B. Dynamic patterns of information flow in complex networks. Nat. Commun. 8, 2181 (2017).

  12. 12.

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

  13. 13.

    Barzel, B. & Biham, O. Quantifying the connectivity of a network: the network correlation function method. Phys. Rev. E 80, 046104 (2009).

  14. 14.

    Crucitti, P., Latora, V. & Marchiori, M. Model for cascading failures in complex networks. Phys. Rev. E 69, 045104 (2004).

  15. 15.

    Dobson, I., Carreras, B. A., Lynch, V. E. & Newman, D. E. Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization. Chaos 17, 026103 (2007).

  16. 16.

    Voit, E. O. Computational Analysis of Biochemical Systems (Cambridge Univ. Press, New York, NY, 2000).

  17. 17.

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

  18. 18.

    Karlebach, G. & Shamir, R. Modelling and analysis of gene regulatory networks. Nat. Rev. 9, 770–780 (2008).

  19. 19.

    Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

  20. 20.

    Opsahl, T. & Panzarasa, P. Clustering in weighted networks. Soc. Networks 31, 155–163 (2009).

  21. 21.

    Eckmann, J.-P., Moses, E. & Sergi, D. Entropy of dialogues creates coherent structures in e-mail traffic. Proc. Natl Acad. Sci. USA 101, 14333–14337 (2004).

  22. 22.

    Ikehara, K. & Clauset, A. Characterizing the structural diversity of complex networks across domains. Preprint at (2017).

  23. 23.

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

  24. 24.

    Rual, J. F. et al. Towards a proteome-scale map of the human protein-protein interaction network. Nature 437, 1173–1178 (2005).

  25. 25.

    Sporns, O., Tononi, G. & Kötter, R. The human connectome: a structural description of the human brain. PLoS Comput. Biol. 1, e42 (2005).

  26. 26.

    Robertson, C. Flowers and Insects: Lists of Visitors of Four Hundred and Fifty-three Flowers (C. Robertson, Carlinville, Il., 1929).

  27. 27.

    Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–958 (2015).

  28. 28.

    Stern, M., Sompolinsky, H. & Abbott, L. F. Dynamics of random neural networks with bistable units. Phys. Rev. E 90, 062710 (2014).

  29. 29.

    Gardiner, C. W. Handbook of Stochastic Methods (Springer-Verlag, Berlin, 2004).

  30. 30.

    Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).

  31. 31.

    Hayes, J. F. & Ganesh Babu, T. V. J. Modeling and Analysis of Telecommunications Networks (John Wiley & Sons, Inc, Hoboken, 2004).

  32. 32.

    Kuramoto, Y. Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, Heidelberg, 1984).

  33. 33.

    Newman, M. E. J. Networks - An Introduction (Oxford Univ. Press, New York, 2010).

  34. 34.

    Schmetterer, L. & Sigmund, K. (eds) Hans Hahn Gesammelte Abhandlungen Band 1/Hans Hahn Collected Works Vol. 1 (Springer, Vienna, Austria, 1995).

  35. 35.

    Cohen, R. & Havlin, S. Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701 (2003).

  36. 36.

    Caldarelli, G. Scale-free Networks: Ccomplex Webs in Nature and Technology (Oxford Univ. Press, New York, 2007).

  37. 37.

    Wilson, K. G. The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773 (1975).

  38. 38.

    Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10, 186–198 (2009).

  39. 39.

    Wai, H.-T., Scaglione, A., Harush, U., Barzel, B. & Leshem, A. RIDS: robust identification of sparse gene regulatory networks from perturbation experiments. Preprint at (2017).

  40. 40.

    Novozhilov, A. S., Karev, G. P. & Koonin, E. V. Biological applications of the theory of birth-and-death processes. Brief. Bioinform. 7, 70–85 (2006).

  41. 41.

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

  42. 42.

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

  43. 43.

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

  44. 44.

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

  45. 45.

    Ravaszi, E. B. & Barabási, A.-L. Hierarchical organization in complex networks. Phys. Rev. E 67, 026112 (2003).

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C.H. thanks the Planning and Budgeting Committee (PBC) of the Council for Higher Education, Israel, and the INSPIRE-Faculty grant (code: IFA17-PH193) for support. This work was supported by the US National Science Foundation-CRISP award no. 1735505, the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Directorate in the Prime Minister’s office, and by a grant from the Ministry of Science & Technology, Israel & Ministry of Foreign Affairs and International Cooperation General Directorate for Country Promotion, Italian Republic.

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All authors designed and conducted the research. C.H. and U.H. analysed the data and performed the numerical simulations. B.B. was the lead principal investigator.

Correspondence to Baruch Barzel.

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The authors declare no competing interests.

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Supplementary Information

Supplementary Methods, Supplementary Tables 1–5, Supplementary Figures 1–17 and Supplementary References 1–36.

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Hens, C., Harush, U., Haber, S. et al. Spatiotemporal signal propagation in complex networks. Nat. Phys. 15, 403–412 (2019) doi:10.1038/s41567-018-0409-0

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