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Controllability of complex networks

Abstract

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.

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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.

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Acknowledgements

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.

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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.

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Correspondence to Albert-László Barabási.

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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)

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Liu, YY., Slotine, JJ. & Barabási, AL. Controllability of complex networks. Nature 473, 167–173 (2011). https://doi.org/10.1038/nature10011

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