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Finding the solution for driving a complex network at the minimum energy cost with a given number of controllers, known as the minimum-cost control problem, is critically important but remains largely open. We propose a projected gradient method to tackle this problem, which works efficiently in both synthetic and real-life networks. The study is then extended to the case where each controller can only be connected to a single network node to have the lowest connection complexity. We obtain the interesting insight that such connections basically avoid high-degree nodes of the network, which is in resonance with recent observations on controllability of complex networks. Our results provide the first technical path to enabling minimum-cost control of complex networks, and contribute new insights to locating the key nodes from a minimum-cost control perspective. Minimum-cost controlThe minimum-cost control problem can be modelled as driving the state vector to the origin in the time interval [0, t f ] with the minimum energy cost [23]

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Complex networks characterize the nature of internal/external interactions in real-world systems including social, economic, biological, ecological, and technological networks. Two issues keep as obstacles to fulfilling control of large-scale networks: structural controllability which describes the ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs; and optimal control, which is a typical control approach to minimize the cost for driving the network to a predefined state with a given number of control inputs. For large complex networks without global information of network topology, both problems remain essentially open. Here we combine graph theory and control theory for tackling the two problems in one go, using only local network topology information. For the structural controllability problem, a distributed local-game matching method is proposed, where every node plays a simple Bayesian game with local information and local interactions with adjacent nodes, ensuring a suboptimal solution at a linear complexity. Starring from any structural controllability solution, a minimizing longest control path method can efficiently reach a good solution for the optimal control in large networks. Our results provide solutions for distributed complex network control and demonstrate a way to link the structural controllability and optimal control together.Over the past decade the complex natural and technological systems that permeate many aspects of everyday life-including human brain intelligence, medical science, social science, biology, and economics-have been widely studied 1-3 . Many of these complex systems can be modeled as static or dynamic networks, which stimulates the emergence and booming developments of research on complex networks. There are two fundamental issues associated with the control of complex networks, with different focuses on figuring out (i) whether the networks are controllable; and (ii) how to control them with least cost when they are controllable, respectively. The first issue is typically investigated by studying the structural controllability problem, which describes the ability to guide a dynamical system from any initial state to any desired final state in finite time. The second issue is known as the optimal cost control problem, with the main objective of minimizing the cost for driving the network to a predefined state with a given number of control inputs. Figure 1 illustrates the structural controllability problem and the optimal cost control problem. Note that for large complex networks without global information of network topology, both problems remain essentially open. In this work, we shall combine graph theory and control theory for tackling the two problems in one go, using only local network topology information.Researchers are using a multidisciplinary approach to study the structural controllability of complex networks, focusing on linear time invariant (LTI) 4 , where x(t) = [x 1 (t), …, ...

Preparation of heteroatom-functionalized porous carbon derived from byproducts of bean curd stick manufacture as an electrode material for high performance supercapacitors.