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1. P. S. Laplace, A Philosophical Essay on Probabilities Théorie Analytique des Probabilités, 6th ed. ( Madame Veuve Courcier, Paris, 1820).
2. D. J. Watts and S. H. Strogatz, “ Collective dynamics of 'small-world' networks,” Nature 393, 440 (1998).
3. A.-L. Barabási and R. Albert, “ Emergence of scaling in random networks,” Science 286(5439), 509 (1999).
4. P. W. Anderson, “ More is different,” Science 177(4047), 393 (1972).
5. B. Ø. Palsson, Systems Biology: Properties of Reconstructed Networks ( Cambridge University Press, Cambridge, United Kingdom, 2006).
6.The network systems currently investigated are also significantly different from well-mixed or well-ordered complex systems (think of a gas, a crystal) to which statistical mechanics has been traditionally applied. They are well structured and as such cannot be characterized by few average quantities; and the equivalent to a unit cell would be essentially of the size of the entire system, which in turn leads to the issue of scale. Complexity is ultimately a statement about the dynamics (rather than the structure) of the system, which is nevertheless influenced by structure in such networks.
7. K. Zhang and T. J. Sejnowski, “ A universal scaling law between gray matter and white matter of cerebral cortex,” Proc. Natl. Acad. Sci. U. S. A. 97, 5621 (2000).
8. J. C. Sanchez and J. C. Principe, Brain-Machine Interface Engineering ( Morgan & Claypool Publishers, New York, NY, 2007).
9. S. J. Schiff, Neural Control Engineering ( MIT Press, Cambridge, MA, 2012).
10.Thus, this article focuses on the control of networks rather than “network control” as used to refer to distributed control systems that use networks of sensors and actuators to control a system that is not necessarily a network.11
11. J. Baillieul and P. J. Antsaklis, “ Control and communication challenges in networked real-time systems,” Proc. IEEE 95(1), 928 (2007).
12. M. E. J. Newman, “ The structure and function of complex networks,” SIAM Rev. 45, 167 (2003).
13. K. J. Astrom and P. R. Kumar, “ Control: A perspective,” Automatica 50, 3 (2014).
14. R. E. Kalman, “ Mathematical description of linear dynamical systems,” J. Soc. Ind. Appl. Math., Ser. A 1, 152 (1963).
15. J. Sun, S. P. Cornelius, W. L. Kath, and A. E. Motter, “ Comment on ‘Controllability of complex networks with nonlinear dynamics’,” e-print arXiv:1108.5739 (2011).
16. G. W. Haynes and H. Hermes, “ Nonlinear controllability via Lie theory,” SIAM J. Control 8, 450 (1970).
17. H. J. Sussmann and V. Jurdjevic, “ Controllability of nonlinear systems,” J. Differ. Equations 12, 95 (1972).
18. R. Hermann and A. J. Krener, “ Nonlinear controllability and observability,” IEEE Trans. Automat. Control 22, 728 (1977).
19. J. C. Maxwell, “ On governors,” Proc. R. Soc. London 16, 270 (1868).
20. J. C. Maxwell, “ On reciprocal figures and diagrams of forces,” Philos. Mag. 27, 250 (1864).
21. S. Meyn, Control Techniques for Complex Networks ( Cambridge University Press, Cambridge, United Kingdom, 2007).
22. F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks ( Princeton University, Princeton, NJ, 2009).
23. H. Su and W. Xiaofan, Pinning Control of Complex Networked Systems ( Springer, Berlin, 2013).
24. D. Lozovanu and S. Pickl, Optimization of Stochastic Discrete Systems and Control on Complex Networks ( Springer, Berlin, 2015).
25. C.-T. Lin, “ Structural controllability,” IEEE Trans. Automat. Control AC-19(3), 201 (1974).
26. R. E. Kalman, “ On the general theory of control systems,” in Proceedings of the First IFAC Congress on Automatic Control, Moscow, 1960 (1961), Vol. 1, pp. 481492.
27. K. Murota, Systems Analysis by Graphs and Matroids: Structural Solvability and Controllability, 1st ed. ( Springer, Berlin, 1987).
28. E. F. Camacho and C. Bordons, Model Predictive Control ( Springer, Berlin, 2013).
29. R. Cohen, S. Havlin, and D. ben-Avraham, “ Efficient immunization strategies for computer networks and populations,” Phys. Rev. Lett. 91, 247901 (2003).
30. S. L. Feld, “ Why your friends have more friends than you do,” Am. J. Sociol. 96, 1464 (1991).
31. N. A. Christakis and J. H. Fowler, “ Social network sensors for early detection of contagious outbreaks,” PLoS One 5(9), e12948 (2010).
32. J. Sun and A. E. Motter, “ Controllability transition and nonlocality in network control,” Phys. Rev. Lett. 110, 208701 (2013).
33. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods ( SIAM, Philadelphia, PA, 1998).
34. C. Commault, J.-M. Dion, and J. W. van der Woude, “ Characterization of generic properties of linear structured systems for efficient computations,” Kybernetika 38, 503 (2002); available at
35. Y. Y. Liu, J. J. Slotine, and A.-L. Barabási, “ Controllability of complex networks,” Nature 473, 167 (2011).
36. B. Friedland, “ Controllability index based on conditioning number,” J. Dyn. Syst., Meas., Control 97, 444 (1975).
37. N. J. Cowan, E. J. Chastain, D. A. Vilhena, J. S. Freudenberg, and C. T. Bergstrom, “ Nodal dynamics, not degree distributions, determine the structural controllability of complex networks,” PLoS One 7, e38398 (2012).
38. J. Sun and A. E. Motter (unpublished).
39. J. Lévine, Analysis and Control of Nonlinear Systems: A Flatness-Based Approach (Springer, Berlin, 2009).
40.Note that the challenge is to steer the trajectory toward a specific different attractor (hence, across different ergodic regions of the phase space). It is not important whether the attractor is a fixed point—it can be periodic or even chaotic since, once the attractor is reached, simple methods can be used to manipulate that dynamics within it.41,42
41. E. M. Bollt, “ Targeting control of chaotic systems,” in Chaos and Bifurcations Control: Theory and Applications, edited by G. Chen, X. Yu, and D. J. Hill ( Springer, Berlin, 2004), pp. 124.
42. E. Ott, C. Grebogi, and J. A. Yorke, “ Controlling chaos,” Phys. Rev. Lett. 64, 1196 (1990).
43. S. P. Cornelius, W. L. Kath, and A. E. Motter, “ Realistic control of network dynamics,” Nat. Commun. 4, 1942 (2013).
44.It goes without saying that the intersection of the feasible region with the target basin of attraction will generally depend on time of the intervention. A dramatic illustration of this dependence was shown in Ref. 45, where examples were given of extinction cascades in food-web networks that could be prevented entirely by the suppression of species that would otherwise be eventually extinct by the cascade.
45. S. Sahasrabudhe and A. E. Motter, “ Rescuing ecosystems from extinction cascades through compensatory perturbations,” Nat. Commun. 2, 170 (2011).
46. S. P. Cornelius and A. E. Motter, “ NECO—A scalable algorithm for NEtwork COntrol,” Protoc. Exchange doi:10.1038/protex.2013.063 (2013).
47.Incidentally, here is where we take advantage of the sparsity that sets networks apart from other dynamical systems. The approach can be applied to any dynamical system at the estimated computational cost of O(n3.5), but for networks this cost is reduced by a full power of n provided that the number of variables is approximately proportional to the number of nodes and that the average degree remains essentially constant, as is the case in many network models.
48. D. Murrugarra, A. Veliz-Cuba, B. Aguilar, S. Arat, and R. Laubenbacher, “Modeling stochasticity and variability in gene regulatory networks,” EURASIP J. Bioinformat. Syst. Biol. 2012, 5 (2012).
49. K. Z. Szalay, R. Nussinov, and P. Csermely, “Attractor structures of signaling networks: Consequences of different conformational barcode dynamics and their relations to network-based drug design,” Mol. Inf. 33, 463 (2014).
50. D. K. Wells, W. L. Kath, and A. E. Motter, “ Control of stochastic and induced switching in biophysical networks,” Phys. Rev. X 5, 031036 (2015).
51. Z. G. Nicolaou and A. E. Motter, “ Mechanical metamaterials with negative compressibility transitions,” Nature Mater. 11, 608 (2012).
52. I. Mezic, Controllability, Integrability and Ergodicity, Lecture Notes in Control and Information Sciences ( Springer, Berlin, 2003), Vol. 289, p. 213.
53. I. Mezic, “ Controllability of hamiltonian systems with drift: Action-angle variables and ergodic partition,” in Proceedings of the 42nd IEEE Conference on Decision and Control (2003), Vol. 3, pp. 25852592.
54. Y. Yang, J. Wang, and A. E. Motter, “ Network observability transitions,” Phys. Rev. Lett. 109, 258701 (2012).
55. Y. Y. Liu, J. J. Slotine, and A.-L. Barabási, “ Observability of complex systems,” Proc. Natl. Acad. Sci. U. S. A. 110, 2460 (2013).
56. A. J. Whalen, S. N. Brennan, T. D. Sauer, and S. J. Schiff, “ Observability and controllability of nonlinear networks: The role of symmetry,” Phys. Rev. X 5, 011005 (2015).
57. P.-A. Noël, C. D. Brummitt, and R. M. D'Souza, “ Controlling self-organizing dynamics on networks using models that self-organized,” Phys. Rev. Lett. 111, 078701 (2013).
58. A. E. Motter, “ Cascade control and defense in complex networks,” Phys. Rev. Lett. 93, 098701 (2004).
59. X. F. Wang and G. Chen, “ Pinning control of scale-free dynamical networks,” Physica A 310(3), 521 (2002).
60. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “ Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
61. A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, “ Spontaneous synchrony in power-grid networks,” Nat. Phys. 9, 191 (2013).
62. Y. Tang, F. Qian, H. Gao, and J. Kurths, “ Synchronization in complex networks and its application—A survey of recent advances and challenges,” Annu. Rev. Control 38, 184 (2014).
63. F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, “ Controllability of complex networks via pinning,” Phys. Rev. E 75, 046103 (2007).
64. M. Porfiri and F. Fiorilli, “ Node-to-node pinning control of complex networks,” Chaos 19, 013122 (2009).
65. W. Yu, G. Chen, J. Lu, and J. Kurths, “ Synchronization via pinning control on general complex networks,” SIAM J. Control Optim. 51(2), 1395 (2013).
66. N. R. Sandell, P. Varaiya, M. Athans, and M. G. Safonov, “ Survey of decentralized control methods for large scale systems,” IEEE Trans. Autom. Control 23, 108 (1978).
67. D. D. Šiljak, Decentralized Control of Complex Systems ( Dover, Mineola, NY, 2012; first published by Academic Press Inc., Boston, MA, 1991).
68. R. D'Andrea and G. E. Dullerud, “ Distributed control design for spatially interconnected systems,” IEEE Trans. Autom. Control 48, 1478 (2003).
69. P. Benner, “ Solving large-scale control problems,” IEEE Control Syst. Mag. 24(1), 44 (2004).
70. D. D. Šiljak and A. I. Zečević, “ Control of large-scale systems: Beyond decentralized feedback,” Annu. Rev. Control 29, 169 (2005).
71. B. Jakubczyk, Introduction to Geometric Nonlinear Control: Controllability and Lie Bracket ( ICTP Lecture Notes, Trieste, 2001).
72. A. Isidori, Nonlinear Control Systems, 3rd ed. ( Springer, Berlin, 1995).

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An increasing number of complex systems are now modeled as networks of coupled dynamical entities. Nonlinearity and high-dimensionality are hallmarks of the dynamics of such networks but have generally been regarded as obstacles to control. Here, I discuss recent advances on mathematical and computational approaches to control high-dimensional nonlinear network dynamics under general constraints on the admissible interventions. I also discuss the potential of network control to address pressing scientific problems in various disciplines.


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