Towards the minimum-cost control of target nodes in directed networks with linear dynamics |
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| Authors: | Leitao Gao Guangshe Zhao Guoqi Li Lei Deng Fei Zeng |
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| Institution: | 1. School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China;2. Qingdao R&D Institute, Xi’an Jiaotong University, Qingdao Shandong, 266300, China;3. Center for Brain Inspired Computing Research, Department of Precision Instrument, Tsinghua University, Beijing 100084, PR China;4. Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA;5. School of Materials Science and Engineering, Tsinghua University, Beijing 100084, P. R. China |
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| Abstract: | Determining an input matrix, i.e., locating predefined number of nodes (named “key nodes”) connected to external control sources that provide control signals, so as to minimize the cost of controlling a preselected subset of nodes (named “target nodes”) in directed networks is an outstanding issue. This problem arises especially in large natural and technological networks. To address this issue, we focus on directed networks with linear dynamics and propose an iterative method, termed as “L0-norm constraint based projected gradient method” (LPGM) in which the input matrix B is involved as a matrix variable. By introducing a chain rule for matrix differentiation, the gradient of the cost function with respect to B can be derived. This allows us to search B by applying probabilistic projection operator between two spaces, i.e., a real valued matrix space RN?×?M and a L0 norm matrix space by restricting the L0 norm of B as a fixed value of M. Then, the nodes that correspond to the M nonzero elements of the obtained input matrix (denoted as ) are selected as M key nodes, and each external control source is connected to a single key node. Simulation examples in real-life networks are presented to verify the potential of the proposed method. An interesting phenomenon we uncovered is that generally the control cost of scale free (SF) networks is higher than Erdos-Renyi (ER) networks using the same number of external control sources to control the same size of target nodes of networks with the same network size and mean degree. This work will deepen the understanding of optimal target control problems and provide new insights to locate key nodes for achieving minimum-cost control of target nodes in directed networks. |
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| Keywords: | Corresponding authors |
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