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Dynamic interdependence and competition in multilayer networks

Nature Physics volume 15, pages 178–185 (2019)Cite this article

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Abstract

From critical infrastructure to physiology and the human brain, complex systems rarely occur in isolation. Instead, the functioning of nodes in one system often promotes or suppresses the functioning of nodes in another. Structural interdependence—that is, when the functionality of the nodes is determined exclusively by connectivity between layers—can be characterized via percolation processes on interdependent networks. However, modelling more general interactions between dynamical systems has remained an open problem. Here, we present a dynamic dependency framework that can capture interdependent and competitive interactions between dynamic systems, which we use to study synchronization and spreading processes in multilayer networks with interacting layers. By developing a mean-field theory, which we verify by simulations, we find coupled collective phenomena, including multistability, regions of coexistence, and macroscopic chaos. In interdependent dynamics, in particular, we observe hysteretic behaviours with abrupt (hybrid and explosive) transitions, that exhibit universal features that match those emerging from interdependent percolation. This dynamic dependency framework provides a powerful tool with which to improve our understanding of many of the interacting complex systems surrounding us.

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Fig. 1: Dynamic interdependence and competition.
Fig. 2: Interdependent synchronization.
Fig. 3: Competitive synchronization.
Fig. 4: Asymmetric synchronization.
Fig. 5: Interacting epidemics.
Fig. 6: Universal scaling in interdependent dynamics.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

M.D. thanks the Azrieli Foundation for the award of an Azrieli Fellowship Grant. S.H. acknowledges the Israel Science Foundation, ONR Global, the Israel-Italian collaborative project NECST, Japan Science Foundation, BSF-NSF, and DTRA (Grant no. HDTRA-1-10-1-0014) for financial support. S.H. also thanks the Nvidia Corporation for a small hardware grant, which was used for this research. This work was supported by the Israeli Ministry of Science, Technology and Space (MOST grant #3-12072). We also thank B. Barzel, A.K.H. Chan and V. Zlatic for several helpful comments and discussions about the manuscript.

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  1. These authors contributed equally to this work: Michael M. Danziger, Ivan Bonamassa

Authors and Affiliations

  1. Department of Physics, Northeastern University, Boston, MA, USA

    Michael M. Danziger

  2. Department of Physics, Bar-Ilan University, Ramat Gan, Israel

    Ivan Bonamassa & Shlomo Havlin

  3. CNR - Institute for Complex Systems, Sesto Fiorentino, Italy

    Stefano Boccaletti

  4. Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi’an, China

    Stefano Boccaletti

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  1. Michael M. Danziger
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Contributions

M.M.D., I.B., S.B. and S.H. developed the concept. I.B. and M.M.D. jointly designed the framework. I.B. developed the analytic results. M.M.D. created all of the simulations and figures. S.B. provided conceptual advice. M.M.D., I.B. and S.H. wrote the paper.

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Correspondence to Michael M. Danziger.

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Danziger, M.M., Bonamassa, I., Boccaletti, S. et al. Dynamic interdependence and competition in multilayer networks. Nature Phys 15, 178–185 (2019). https://doi.org/10.1038/s41567-018-0343-1

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