Spectrum of controlling and observing complex networks

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.

References

  1. 1

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. 2

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

    Google Scholar 

  3. 3

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

    Google Scholar 

  4. 4

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. 5

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

    Google Scholar 

  6. 6

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

    ADS  Article  Google Scholar 

  7. 7

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

    Google Scholar 

  8. 8

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

    Google Scholar 

  9. 9

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

    Google Scholar 

  10. 10

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

    ADS  Article  Google Scholar 

  11. 11

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

    ADS  Article  Google Scholar 

  12. 12

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

    ADS  Article  Google Scholar 

  13. 13

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

    MathSciNet  MATH  Article  Google Scholar 

  14. 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).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. 15

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

    ADS  Article  Google Scholar 

  16. 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).

    ADS  Article  Google Scholar 

  17. 17

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

    ADS  Article  Google Scholar 

  18. 18

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

    MathSciNet  MATH  Article  Google Scholar 

  19. 19

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

    ADS  Article  Google Scholar 

  20. 20

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

    ADS  Article  Google Scholar 

  21. 21

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 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).

    ADS  Article  Google Scholar 

  23. 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. 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. 25

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

    ADS  Article  Google Scholar 

  26. 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. 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. 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).

    MathSciNet  MATH  Article  Google Scholar 

  29. 29

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

    MathSciNet  MATH  Article  Google Scholar 

  30. 30

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31

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

    ADS  Article  Google Scholar 

  32. 32

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

    ADS  Article  Google Scholar 

  33. 33

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

    ADS  Article  Google Scholar 

  34. 34

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

    ADS  Article  Google Scholar 

  35. 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).

    ADS  Article  Google Scholar 

  36. 36

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

    ADS  Article  Google Scholar 

  37. 37

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

    Google Scholar 

  38. 38

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

    ADS  Article  Google Scholar 

  39. 39

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. 40

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. 41

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

    ADS  Article  Google Scholar 

  42. 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).

    ADS  Article  Google Scholar 

  43. 43

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

    Google Scholar 

  44. 44

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

    ADS  Article  Google Scholar 

  45. 45

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

    Google Scholar 

  46. 46

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

    ADS  MATH  Article  Google Scholar 

  47. 47

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

    ADS  Article  Google Scholar 

  48. 48

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. 49

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

    ADS  Article  Google Scholar 

  50. 50

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

    ADS  Article  Google Scholar 

  51. 51

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

    Article  Google Scholar 

  52. 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. 53

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

    Google Scholar 

  54. 54

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

    ADS  Article  Google Scholar 

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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).

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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.

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Correspondence to Albert-László Barabási.

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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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