Finding graph minimum stable set and core via semi-tensor product approach
Introduction
The development of graph theory was largely inspired and guided by the famous conjecture called Four-Colour Conjecture. The conjecture was solved by Appel and Haken in 1976 [1], which marked a big turning point in the history of graph theory. Since then, graph theory has experienced rapid and explosive growth, due to its important role in the applied mathematics [2], [3], [4], [5]. With the rapid development of computer science and combinational optimization [6], [7], the versatility of graph makes them indispensable tools to analyze and design the large-scale communication networks [8].
Recently, graph theory has also been widely used in the systematical analysis of neural networks, complex networks, and multi-agent systems, which are some of the hottest topics in the control field. For example, the Laplacian matrix of graph plays an extremely important role in control protocol designs for multi-agent with linear dynamics and synchronization of complex networks. Many fundamental and landmark results have been obtained regarding the Laplacian matrix [9]. A spanning tree T of an undirected graph is a subgraph that includes all of the vertices of . The importance of spanning trees of various special types has been evident, which are widely used in the analysis of multi-agent systems.
The minimum (externally) stable set is another basic and classical problem in graph theory, which has wide applications in competitive markets, stochastic systems, clustering and so on [10], [11], [12], [13]. Externally stable set requires that each node which is not belonging to the set is connected at least one node belonging to the set [14]. For example, externally stable set can be used to analyze the stability of 2-D systems [15]. As an opposite problem of graph, the investigation of internally stable set has been studied in [16], which can be seen as an independent set, or a vertex packing of graph. Externally stable set and internally stable set are two important and independent properties in graph theory.
Recently, a new powerful matrix product, called semi-tensor product of matrices was proposed by Cheng and his colleagues [17], [18]. This new matrix product provides a way to multiply two matrices with arbitrary dimensions [17]. By resorting to semi-tensor product, a Boolean function can be converted into an algebraic form, and then a Boolean network can be expressed as a discrete algebraic system [17]. This original set-up opens new perspectives on systematical analysis of many problems for Boolean networks. And up to now, many fundamental and landmark results have been presented on calculating fixed points and cycles of Boolean networks [17], [19], on the controllability and observability of Boolean networks [20], [21], [22], [23], on the stability of Boolean networks [24], on the optimal control of Boolean control networks [25], on the synchronization of Boolean networks [26], [27], [28], [29], [30], [31], [32], on graph coloring problems [16], on Kalman decomposition of Boolean control networks [33], on networked evolutionary games [34], on nonlinear feedback shift registers [35] and so on.
It should be noted that one main drawback of the algebraic state expression of Boolean networks is its computational complexity. The algebraic state representation converts a Boolean network with n state-variables into a state-space of size 2n. Thus, any algorithm based on this approach has an exponential time-complexity. Moreover, many problems like determining fixed points and observability of Boolean control networks have already been proved to be NP-hard. Hence, the computational complexity is intrinsic and also independent of the models adopted to describe Boolean networks. The main contributions of this paper are as follows: (i) some algebraic descriptions have been established to deal with the minimum stable set and graph core; (ii) a set of theoretical results and algorithms have been presented to determine the minimum stable set and graph core.
The rest of this paper is structured as follows. In Section 2, we present some preliminaries on semi-tensor product, k-valued logical variables, pseudo-Boolean function and graph theory, and definitions of (absolutely minimum) externally stable set and (absolutely maximum) internally stable set. In Section 3, we investigate (absolutely minimum) externally stable set and graph core, and then obtain some necessary and sufficient conditions and efficient algorithms to find the externally stable sets and graph cores. The study of numerical examples shows that the obtained results/algorithms are very effectiveness. The conclusion is presented in Section 4.
Notations: The standard notations will be used in this paper. Throughout this paper, denotes the set of real matrices of order n×m, and denotes the positive integers. denotes the n-dimensional column vector with all entries being 1, and Ik is the identity matrix of order k. δkj is the j-th column of identity matrix Ik, and Δk denotes the set of all k columns of Ik. Let be the j-th column of matrix A, and be the set of columns of matrix A.
Section snippets
Some preliminaries
In this section, we give an outline of semi-tensor product of matrices, k-valued logical variables, pseudo-Boolean function and graph theory, which will be used in the following sequels.
Now we are in the position to give the definition of semi-tensor product of matrices. Definition 1 For a n×m matrix A and a p×q matrix B. Let l be the least common multiple of m and p. Then the semi-tensor product of A and B is defined as follows: Cheng et al. [18]
Here is the Kronecker product of matrices. We can see
Main results
Consider a graph with n vertices . Assume that the adjacency matrix of , , is given asThen, based on the adjacency matrix , we define a new matrix, called unitary association matrix, , of , which is given asHence, according to Eq. (6), we can obtain that , where δij is the Kronecker sign, and defined by
For example, consider three graphs, as shown in Fig. 1. We can obtain the corresponding
Conclusion
In this paper, we have investigated the minimum stable set and graph core. A number of theoretical results and algorithms are derived to find the minimum stable set and graph core. Using the semi-tensor product of matrices and the characteristic logical vector, we obtained the algebraic descriptions for the minimum stable set and graph core. Based on the algebraic representations, some necessary and sufficient conditions are obtained to determine the minimum stable set and graph core. According
Acknowledgements
This work was partially supported by the NNSF of China under Grants 61175119, 61272530, 61573102 and 61573096, and China Postdoctoral Science Foundation under Grant 2014M560377.
Jie Zhong received his B.S. degree from Zhejiang Normal University in Zhejiang, China, in 2012, the M.S. degree from Southeast University in Nanjing, China, in 2014. He is currently pursuing his Ph.D. degree from the City University of Hong Kong, Hong Kong. His research interests include Boolean (control) networks and complex networks.
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Jie Zhong received his B.S. degree from Zhejiang Normal University in Zhejiang, China, in 2012, the M.S. degree from Southeast University in Nanjing, China, in 2014. He is currently pursuing his Ph.D. degree from the City University of Hong Kong, Hong Kong. His research interests include Boolean (control) networks and complex networks.
Jianquan Lu received the B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2003, the M.S. degree in mathematics from Southeast University, Nanjing, China, in 2006, and the Ph.D. degree from City University of Hong Kong, Hong Kong, in 2009. He is currently a professor at the Department of Mathematics, Southeast University, Nanjing, China. His current research interests include collective behavior in complex dynamical networks and multi-agent systems, and Boolean control networks. He has published over 40 papers in refereed international journals.
Dr. Lu is an associate editor of Neural Processing Letters (Springer), Neural Computing and Applications (Springer), and a guest editor of Mathematics and Computers in Simulation (Elsevier). He was elected Most Cited Chinese Researchers by Elsevier, the recipient of an Alexander von Humboldt Fellowship in 2010, Program for New Century Excellent Talents in University by The Ministry of Education, China, in 2010, and The First Award of Jiangsu Provincial Progress in Science and Technology in 2010 as the Second Project Member.
Chi Huang received the M.S. and the Ph.D. degree in mathematics from City University of Hong Kong, Kowloon, Hong Kong, in 2009 and 2013, respectively. He is currently an associate professor at the College of Mathematics, Taiyuan University of Technology, Taiyuan, China. His research interests include multi-agent systems, complex dynamical networks and sensor networks. He has published over 20 academic papers.
Dr. Huang was the recipient of the Best Paper Award from the 8th Asian Control Conference (ASCC2011).
Lulu Li received the B.S. degree in mathematics and applied mathematics from Anhui Normal University, Wuhu, China, in 2007 and the M.S. degree in mathematics from Southeast University, Nanjing, China, in 2010 and the Ph.D. degree from City University of Hong Kong, Hong Kong, in 2013. He is currently an associate professor at the School of Mathematics, Hefei University of Technology, Hefei 230009, China. His current research interests include nonlinear system, collective behavior in complex dynamical networks and multi-agent systems, and stability theory.
Jinde Cao received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989 and 1998, respectively. From March 1989 to May 2000, he was with the Yunnan University. In May 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. From July 2001 to June 2002, he was a Postdoctoral Research Fellow at the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Hong Kong. In the period from 2006 to 2008, he was a Visiting Research Fellow and a Visiting Professor at the School of Information Systems, Computing and Mathematics, Brunel University, UK. On August 2014, he was a Visiting Professor at the School of Electrical and Computer Engineering, RMIT University, Australia. Currently, he is a distinguished professor and doctoral advisor at the Southeast University, prior to which he was a Professor at Yunnan University from 1996 to 2000. His research interests include nonlinear systems, neural networks, complex systems and complex networks, stability theory, and applied mathematics. He was an Associate Editor of the IEEE Transactions on Neural Networks, Journal of the Franklin Institute and Neurocomputing. He is an Associate Editor of the IEEE Transactions on Cybernetics, Differential Equations and Dynamical Systems, Mathematics and Computers in Simulation, and Neural Networks. He is a Reviewer of Mathematical Reviews and Zentralblatt-Math. He is a ISI Highly Cited Researcher in Mathematics and Engineering listed by Thomson Reuters.