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Spectrum of controlling and observing complex networks

Nature Physics volume 11, pages 779–786 (2015)Cite this article

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Abstract

Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. But the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for different directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.

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Figure 1: Controlling and observing a network.
Figure 2: Controlling a network through all nodes (ND = N).
Figure 3: Controlling a network through a single node (ND = 1).
Figure 4: Controlling a network through a finite fraction of its nodes.

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Acknowledgements

We thank E. Guney, C. Song, J. Gao, M. T. Angulo, S. P. Cornelius, B. Coutinho and A. Li for discussions. This work was supported by Army Research Laboratories (ARL) Network Science (NS) Collaborative Technology Alliance (CTA) grant ARL NS-CTA W911NF-09-2-0053; DARPA Social Media in Strategic Communications project under agreement number W911NF-12-C-002; the John Templeton Foundation: Mathematical and Physical Sciences grant number PFI-777; European Commission grant numbers FP7 317532 (MULTIPLEX) and 641191 (CIMPLEX).

Author information

Author notes

  1. Gang Yan and Georgios Tsekenis: These authors contributed equally to this work.

Authors and Affiliations

  1. Center for Complex Network Research and Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA

    Gang Yan, Georgios Tsekenis & Albert-László Barabási

  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

    Baruch Barzel

  3. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    Jean-Jacques Slotine

  4. Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    Jean-Jacques Slotine

  5. Channing Division of Network Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA

    Yang-Yu Liu

  6. Center for Cancer Systems Biology, Dana Farber Cancer Institute, Boston, Massachusetts 02115, USA

    Yang-Yu Liu & Albert-László Barabási

  7. Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA

    Albert-László Barabási

  8. Center for Network Science, Central European University, H-1051 Budapest, Hungary

    Albert-László Barabási

Authors
  1. Gang Yan
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Contributions

All authors designed and performed the research. G.Y. and G.T. carried out the numerical calculations. G.Y. did the analytical calculations and analysed the empirical data. G.T., B.B., J.-J.S., Y.-Y.L. and A.-L.B. analysed the results. G.Y. and A.-L.B. were the main writers of the manuscript. G.T., B.B. and Y.-Y.L. edited the manuscript. G.Y. and G.T. contributed equally to this work.

Corresponding author

Correspondence to Albert-László Barabási.

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The authors declare no competing financial interests.

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Yan, G., Tsekenis, G., Barzel, B. et al. Spectrum of controlling and observing complex networks. Nature Phys 11, 779–786 (2015). https://doi.org/10.1038/nphys3422

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