Skip to main content

Thank you for visiting 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.

Controllability of complex networks


The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system’s entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network’s degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Controlling a simple network.
Figure 2: Characterizing and predicting the driver nodes ( ND).
Figure 3: The impact of network structure on the number of driver nodes.
Figure 4: Link categories for robust control.
Figure 5: Control robustness.


  1. 1

    Kalman, R. E. Mathematical description of linear dynamical systems. J. Soc. Indus. Appl. Math. Ser. A 1, 152–192 (1963)

    MathSciNet  Article  Google Scholar 

  2. 2

    Luenberger, D. G. Introduction to Dynamic Systems: Theory, Models, & Applications (Wiley, 1979)

    Google Scholar 

  3. 3

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

    Google Scholar 

  4. 4

    Kelly, F. P., Maulloo, A. K. & Tan, D. K. H. Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49, 237–252 (1998)

    Article  Google Scholar 

  5. 5

    Srikant, R. The Mathematics of Internet Congestion Control (Birkhäuser, 2004)

    Google Scholar 

  6. 6

    Chiang, M., Low, S. H., Calderbank, A. R. & Doyle, J. C. Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 95, 255–312 (2007)

    Article  Google Scholar 

  7. 7

    Wang, X. F. & Chen, G. Pinning control of scale-free dynamical networks. Physica A 310, 521–531 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8

    Wang, W. & Slotine, J.-J. E. On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92, 38–53 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9

    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 

  10. 10

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

    MathSciNet  Article  Google Scholar 

  11. 11

    Marucci, L. et al. How to turn a genetic circuit into a synthetic tunable oscillator, or a bistable switch. PLoS ONE 4, e8083 (2009)

    ADS  Article  Google Scholar 

  12. 12

    Strogatz, S. H. Exploring complex networks. Nature 410, 268–276 (2001)

    ADS  CAS  Article  Google Scholar 

  13. 13

    Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of Networks: From Biological Nets to the Internet and WWW. (Oxford Univ. Press, 2003)

    Google Scholar 

  14. 14

    Newman, M., Barabási, A.-L. & Watts, D. J. The Structure and Dynamics of Networks (Princeton Univ. Press, 2006)

    Google Scholar 

  15. 15

    Caldarelli, G. Scale-Free Networks: Complex Webs in Nature and Technology (Oxford Univ. Press, 2007)

    Google Scholar 

  16. 16

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

    ADS  MathSciNet  Article  Google Scholar 

  17. 17

    Tanner, H. G. in Proc. 43rd IEEE Conf. Decision Contr. Vol. 3. 2467–2472 (2004)

    Google Scholar 

  18. 18

    Lombardi, A. & Hörnquist, M. Controllability analysis of networks. Phys. Rev. E 75, 56110 (2007)

    ADS  Article  Google Scholar 

  19. 19

    Liu, B., Chu, T., Wang, L. & Xie, G. Controllability of a leader–follower dynamic network with switching topology. IEEE Trans. Automat. Contr. 53, 1009–1013 (2008)

    MathSciNet  Article  Google Scholar 

  20. 20

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

    MathSciNet  Article  Google Scholar 

  21. 21

    Kim, D.-H. & Motter, A. E. Slave nodes and the controllability of metabolic networks. N. J. Phys. 11, 113047 (2009)

    Article  Google Scholar 

  22. 22

    Mesbahi, M. & Egerstedt, M. Graph Theoretic Methods in Multiagent Networks (Princeton Univ. Press, 2010)

    Google Scholar 

  23. 23

    Motter, A. E., Gulbahce, N., Almaas, E. & Barabási, A.-L. Predicting synthetic rescues in metabolic networks. Mol. Syst. Biol. 4, 168 (2008)

    Article  Google Scholar 

  24. 24

    Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge Univ. Press, 2004)

    Google Scholar 

  25. 25

    Lezon, T. R., Banavar, J. R., Cieplak, M., Maritan, A. & Fedoroff, N. V. Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns. Proc. Natl Acad. Sci. USA 103, 19033–19038 (2006)

    ADS  CAS  Article  Google Scholar 

  26. 26

    Lin, C.-T. Structural controllability. IEEE Trans. Automat. Contr. 19, 201–208 (1974)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27

    Shields, R. W. & Pearson, J. B. Structural controllability of multi-input linear systems. IEEE Trans. Automat. Contr. 21, 203–212 (1976)

    Article  Google Scholar 

  28. 28

    Lohmiller, W. & Slotine, J.-J. E. On contraction analysis for nonlinear systems. Automatica 34, 683–696 (1998)

    MathSciNet  Article  Google Scholar 

  29. 29

    Yu, W., Chen, G., Cao, M. & Kurths, J. Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man Cybern. B 40, 881–891 (2010)

    Article  Google Scholar 

  30. 30

    Hopcroft, J. E. & Karp, R. M. An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)

    MathSciNet  Article  Google Scholar 

  31. 31

    Albert, R., Jeong, H. & Barabási, A.-L. Error and attack tolerance of complex networks. Nature 406, 378–382 (2000)

    ADS  CAS  Article  Google Scholar 

  32. 32

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

    ADS  CAS  Article  Google Scholar 

  33. 33

    Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)

    ADS  CAS  Article  Google Scholar 

  34. 34

    Nishikawa, T., Motter, A. E., Lai, Y.-C. & Hoppensteadt, F. C. Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003)

    ADS  Article  Google Scholar 

  35. 35

    Erdős, P. & Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–60 (1960)

    MathSciNet  MATH  Google Scholar 

  36. 36

    Bollobás, B. Random Graphs (Cambridge Univ. Press, 2001)

    Google Scholar 

  37. 37

    Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  38. 38

    Goh, K.-I., Kahng, B. & Kim, D. Universal behavior of load distribution in scale-free networks. Phys. Rev. Lett. 87, 278701 (2001)

    CAS  Article  Google Scholar 

  39. 39

    Chung, F. & Lu, L. Connected component in random graphs with given expected degree sequences. Ann. Combin. 6, 125–145 (2002)

    MathSciNet  Article  Google Scholar 

  40. 40

    Maslov, S. & Sneppen, K. Specificity and stability in topology of protein networks. Science 296, 910–913 (2002)

    ADS  CAS  Article  Google Scholar 

  41. 41

    Milo, R. et al. Network motifs: simple building blocks of complex networks. Science 298, 824–827 (2002)

    ADS  CAS  Article  Google Scholar 

  42. 42

    Mézard, M. & Parisi, G. The Bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217–233 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  43. 43

    Zdeborová, L. & Mézard, M. The number of matchings in random graphs. J. Stat. Mech. 05, 05003 (2006)

    MathSciNet  Article  Google Scholar 

  44. 44

    Zhou, H. & Ou-Yang, Z.-c. Maximum matching on random graphs. Preprint at 〈〉 (2003)

  45. 45

    Callaway, D. S., Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471 (2000)

    ADS  CAS  Article  Google Scholar 

  46. 46

    Boguñá, M., Pastor-Satorras, R. & Vespignani, A. Cut-offs and finite size effects in scale-free networks. Eur. Phys. J. B 38, 205–209 (2004)

    ADS  Article  Google Scholar 

  47. 47

    Lovász, L. & Plummer, M. D. Matching Theory (American Mathematical Society, 2009)

    Google Scholar 

  48. 48

    Bauer, M. & Golinelli, O. Core percolation in random graphs: a critical phenomena analysis. Eur. Phys. J. B 24, 339–352 (2001)

    ADS  CAS  Article  Google Scholar 

  49. 49

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

    ADS  CAS  Article  Google Scholar 

  50. 50

    Pastor-Satorras, R., Vázquez, A. & Vespignani, A. Dynamical and correlation properties of the Internet. Phys. Rev. Lett. 87, 258701 (2001)

    ADS  CAS  Article  Google Scholar 

Download references


We thank C. Song, G. Bianconi, H. Zhou, L. Vepstas, N. Gulbahce, H. Jeong, Y.-Y. Ahn, B. Barzel, N. Blumm, D. Wang, Z. Qu and Y. Li for discussions. This work was supported by the Network Science Collaborative Technology Alliance sponsored by the US Army Research Laboratory under Agreement Number W911NF-09-2-0053; the Office of Naval Research under Agreement Number N000141010968; the Defense Threat Reduction Agency awards WMD BRBAA07-J-2-0035 and BRBAA08-Per4-C-2-0033; and the James S. McDonnell Foundation 21st Century Initiative in Studying Complex Systems.

Author information




All authors designed and did the research. Y.-Y.L. analysed the empirical data and did the analytical and numerical calculations. A.-L.B. was the lead writer of the manuscript.

Corresponding author

Correspondence to Albert-László Barabási.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data comprising: 1 Introduction; II Previous Work and Relation of Controllability to Other Problems; III Structural Control Theory; IV Maximum Matching; V Control Robustness and VI Network Datasets (see Contents list for full details), Supplementary Figures 1-11 with legends, Supplementary Table 1 and additional references. (PDF 972 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liu, YY., Slotine, JJ. & Barabási, AL. Controllability of complex networks. Nature 473, 167–173 (2011).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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