Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Spectrum of controlling and observing complex networks

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

Subjects

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.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

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.

References

  1. Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Cohen, R. & Havlin, S. Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, 2010).

    Book  MATH  Google Scholar 

  3. Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010).

    Book  MATH  Google Scholar 

  4. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. Rugh, W. J. Linear System Theory (Prentice-Hall, 1996).

    MATH  Google Scholar 

  8. Sontag, E. D. Mathematical Control Theory: Deterministic Finite Dimensional Systems (Springer, 1996).

    Google Scholar 

  9. Slotine, J.-J. & Li, W. Applied Nonlinear Control (Prentice-Hall, 1991).

    MATH  Google Scholar 

  10. Liu, Y.-Y., Slotine, J.-J. & Barabási, A.-L. Controllability of complex networks. Nature 473, 167–173 (2011).

    Article  ADS  Google Scholar 

  11. Yuan, Z., Zhao, C., Di, Z., Wang, W.-X. & Lai, Y.-C. Exact controllability of complex networks. Nature Commun. 4, 2447 (2013).

    Article  ADS  Google Scholar 

  12. Sorrentino, F., di Bernardo, M., Garofalo, F. & Chen, G. Controllability of complex networks via pinning. Phys. Rev. E 75, 046103 (2007).

    Article  ADS  Google Scholar 

  13. Yu, W., Chen, G. & Lü, J. On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  14. Rajapakse, I., Groudine, M. & Mesbahi, M. Dynamics and control of state-dependent networks for probing genomic organization. Proc. Natl Acad. Sci. USA 108, 17257–17262 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Nepusz, T. & Vicsek, T. Controlling edge dynamics in complex networks. Nature Phys. 8, 568–573 (2012).

    Article  ADS  Google Scholar 

  16. Yan, G., Ren, J., Lai, Y.-C., Lai, C.-H. & Li, B. Controlling complex networks: How much energy is needed? Phys. Rev. Lett. 108, 218703 (2012).

    Article  ADS  Google Scholar 

  17. Sun, J. & Motter, A. E. Controllability transition and nonlocality in network control. Phys. Rev. Lett. 110, 208701 (2013).

    Article  ADS  Google Scholar 

  18. Pasqualetti, F., Zampieri, S. & Bullo, F. Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst. 1, 40–52 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. Tang, Y., Gao, H., Zou, W. & Kurths, J. Identifying controlling nodes in neuronal networks in different scales. PLoS ONE 7, e41375 (2012).

    Article  ADS  Google Scholar 

  20. Jia, T. et al. Emergence of bimodality in controlling complex networks. Nature Commun. 4, 2002 (2013).

    Article  ADS  Google Scholar 

  21. Ruths, J. & Ruths, D. Control profiles of complex networks. Science 343, 1373–1376 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Menichetti, G., Dall’Asta, L. & Bianconi, G. Network controllability is determined by the density of low in-degree and out-degree nodes. Phys. Rev. Lett. 113, 078701 (2014).

    Article  ADS  Google Scholar 

  23. Summers, T. H., Cortesi, F. L. & Lygeros, J. On submodularity and controllability in complex dynamical networks. Preprint at http://arXiv.org/abs/1404.7665v2 (2014)

  24. Tzoumas, V., Rahimian, M. A., Pappas, G. J. & Jadbabaie, A. Minimal actuator placement with optimal control constraints. Preprint at http://arXiv.org/abs/1503.04693 (2015)

  25. Cornelius, S. P., Kath, W. L. & Motter, A. E. Realistic control of network dynamics. Nature Commun. 4, 1942 (2013).

    Article  ADS  Google Scholar 

  26. Whalen, A. J., Brennan, S. N., Sauer, T. D. & Schiff, S. J. Observability and controllability of nonlinear networks: The role of symmetry. Phys. Rev. X 5, 011005 (2015).

    Google Scholar 

  27. Menolascina, F. et al. In-vivo real-time control of protein expression from endogenous and synthetic gene networks. PLoS Comput. Biol. 10, e1003625 (2014).

    Article  Google Scholar 

  28. Rahmani, A., Ji, M., Mesbahi, M. & Egerstedt, M. Controllability of multi-agent systems from a graph-theoretic perspective. SIAM J. Control Optim. 48, 162–186 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  29. Acemoglu, D., Ozdaglar, A. & ParandehGheibi, A. Spread of (mis)information in social networks. Games Econ. Behav. 70, 194–227 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Y.-Y., Slotine, J.-J. & Barabási, A.-L. Observability of complex systems. Proc. Natl Acad. Sci. USA 110, 2460–2465 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Yang, Y., Wang, J. & Motter, A. E. Network observability transitions. Phys. Rev. Lett. 109, 258701 (2012).

    Article  ADS  Google Scholar 

  32. Pinto, P. C., Thiran, P. & Vetterli, M. Locating the source of diffusion in large-scale networks. Phys. Rev. Lett. 109, 068702 (2012).

    Article  ADS  Google Scholar 

  33. Scheffer, M. et al. Anticipating critical transitions. Science 338, 344–348 (2012).

    Article  ADS  Google Scholar 

  34. Friedman, N. Inferring cellular networks using probabilistic graphical models. Science 303, 799–805 (2004).

    Article  ADS  Google Scholar 

  35. Almaas, E., Kovács, B., Vicsek, T., Oltvai, Z. N. & Barabási, A.-L. Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature 427, 839–843 (2004).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  37. May, R. M. Stability and Complexity in Model Ecosystems (Princeton Univ. Press, 1974).

    Google Scholar 

  38. Pecora, L. M. & Carroll, T. L. Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998).

    Article  ADS  Google Scholar 

  39. Chung, F., Lu, L. & Vu, V. Spectra of random graphs with given expected degrees. Proc. Natl Acad. Sci. USA 100, 6313–6318 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Kim, D. & Kahng, B. Spectral densities of scale-free networks. Chaos 17, 026115 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Cohen, R., Erez, K., ben Avraham, D. & Havlin, S. Resilience of the Internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000).

    Article  ADS  Google Scholar 

  42. Cowan, N. J., Chastain, E. J., Vilhena, D. A., Freudenberg, J. S. & Bergstrom, C. T. Nodal dynamics, not degree distributions, determine the structural controllability of complex networks. PLoS ONE 7, e38398 (2012).

    Article  ADS  Google Scholar 

  43. Antoulas, A. Approximation of Large-Scale Dynamical Systems (SIAM, 2009).

    MATH  Google Scholar 

  44. Del Genio, C., Gross, T. & Bassler, K. All scale-free networks are sparse. Phys. Rev. Lett. 107, 178701 (2011).

    Article  ADS  Google Scholar 

  45. Kailath, T., Sayed, A. & Hassibi, B. Linear Estimation (Prentice-Hall, 2000).

    MATH  Google Scholar 

  46. Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    Article  ADS  MATH  Google Scholar 

  47. Newman, M. E. J. Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002).

    Article  ADS  Google Scholar 

  48. Girvan, M. & Newman, M. E. J. Community structure in social and biological networks. Proc. Natl Acad. Sci. USA 99, 7821–7826 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Xulvi-Brunet, R. & Sokolov, I. M. Reshuffling scale-free networks: From random to assortative. Phys. Rev. E 70, 066102 (2004).

    Article  ADS  Google Scholar 

  50. Menck, P. J., Heitzig, J., Kurths, J. & Schellnhuber, H. J. How dead ends undermine power grid stability. Nature Commun. 5, 3969 (2014).

    Article  ADS  Google Scholar 

  51. Müller, F.-J. & Schuppert, A. Few inputs can reprogram biological networks. Nature 478, E4 (2011).

    Article  Google Scholar 

  52. Todorov, E. & Jordan, M. I. Optimal feedback control as a theory of motor coordination. Nature Neurosci. 5, 1226–1235 (2002).

    Article  Google Scholar 

  53. Coron, J.-M. Control and Nonlinearity (American Mathematical Society, 2009).

    Book  Google Scholar 

  54. Menck, P. J., Heitzig, J., Marwan, N. & Kurths, J. How basin stability complements the linear-stability paradigm. Nature Phys. 9, 89–92 (2013).

    Article  ADS  Google Scholar 

Download references

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgios Tsekenis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Baruch Barzel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jean-Jacques Slotine
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yang-Yu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Albert-László Barabási
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 1507 kb)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3422

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing