The paradox of controlling complex networks: control inputs versus energy requirement

Yu-Zhong Chen, Lezhi Wang, Wenxu Wang, and Ying-Cheng Lai 3, ∗ School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA Department of Systems Science, Beijing Normal University, Beijing, 10085, China Department of Physics, Arizona State University, Tempe, AZ 85287, USA Abstract One of the most challenging problems in complex dynamical systems is to control complex networks. In previous frameworks based on the structural or the exact controllability theories, the ability to steer a complex network toward any desired state is measured by the minimum number of required driver nodes. However, if we implement actual control by imposing input signals on the minimum set of driver nodes as determined, e.g., by the structural controllability theory, an unexpected phenomenon arises: the energy required to approach a target state with reasonable precision is often unbearably large, precluding us from achieving actual control, i.e., the designated state can not be reached in effect, especially for networks with a small number of drivers. In particular, the energy of controlling a set of networks with similar structural properties follows a fat-tail distribution, indicating the existence of networks with practically divergent energy. We aim to reconcile the paradox of controlling complex networks: optimal structural controllability versus unrealistic energy required for control. We identify fundamental structural “short boards” in complex networks that play a dominant role in the enormous energy, and offer a theoretical interpretation for the fat-tail energy distribution and simple strategies to significantly reduce the energy by imposing slightly augmented set of input signals on properly chosen nodes. Our findings indicate that, although full control can be guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and the control energy to achieve actual control, and our results provide a framework to address this outstanding issue. Notes on the submission history of this work: This work started in late 2012. The phenomena of power-law energy scaling and energy divergence with a single controller were discovered in 2013. Strategies to reduce and optimize control energy was articulated and tested in Spring 2014. The senior co-author (YCL) gave talks about these results at several conferences, including the NETSCI 2014 Satellite entitled “Controlling Complex Networks” on June 2. The paper was submitted to PNAS in September 2014 and was turned down. It was revised and submitted to PRX in early 2015 and was rejected. After that it was revised and submitted to Nature Communications in May 2015 and again was turned down.