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Spatiotemporal signal propagation in complex networks


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.


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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.

Corresponding author

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).

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