Node-set importance and optimization algorithm of nodes selection in complex networks based on pinning control
Authors: Liu Hui , Wang Bing-Jun , Lu Jun-An , Li Zeng-Yang ,   School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China,   School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China,   School of Computer Science, Central China Normal University, Wuhan 430079, China
Published: February 2021

Controlling a complex network to achieve a certain desired objective is an important task for various interacting systems. In many practical situations, it is expensive and unrealistic to control all nodes especially in a large-scale complex network. In order to reduce control cost, one turns to control a small part of nodes in the network, which is called pinning control. This research direction has been widely concerned and much representative progress has been achieved so far. However, to achieve an optimal performance, two key questions about the node-selection scheme remain open. One is how many nodes need controlling and the other is which nodes the controllers should be applied to. It has been revealed in our recent work that the effectiveness of node-selection scheme can be evaluated by the smallest eigenvalue <inline-formula><tex-math id="M1">\begin{document}$ {\rm{\lambda }}_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M1.png"/></alternatives></inline-formula> of the grounded Laplacian matrix obtained by deleting the rows and columns corresponding to the pinned nodes from the Laplacian matrix of the network. As a further study of our previous work, we study node selection algorithm for optimizing pinning control in depth, based on the proposed index <inline-formula><tex-math id="M2">\begin{document}$ {\rm{\lambda }}_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M2.png"/></alternatives></inline-formula> and its spectral properties. As is well known, it is an NP-hard problem to obtain the maximum of <inline-formula><tex-math id="M3">\begin{document}$ {\rm{\lambda }}_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M3.png"/></alternatives></inline-formula> by numerical calculations when the number of pinned nodes is given. To solve this challenge problem, in this paper a filtering algorithm is proposed to find most important nodes, which results in an optimal <inline-formula><tex-math id="M4">\begin{document}$ {\rm{\lambda }}_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20200872_M4.png"/></alternatives></inline-formula> when the number of pinned nodes is given. The method can be applied to any type of undirected networks. Furthermore, in this paper we propose the concept of node-set importance in complex networks from the perspective of network control, which is different from the existing definitions about node importance of complex networks: The importance of a node set and the selected nodes in this paper depends on the number of pinned nodes; if the number of pinned nodes is different, the selected nodes will be different. The concept of node-set importance reflects the effect of nodes’ combination in a network. It is expected that the obtained results are helpful in guiding the optimal control problems in practical networks.