Algebraic criteria for structure identification and behaviour analysis of signed networks

ABSTRACT

This paper deals with the problem on how to algebraically identify structure properties and characterise dynamic behaviours of signed networks with cooperative and antagonistic interactions. A signed digraph is employed to represent the two competitive classes of interactions, and an algebraic criterion based on the determinant of its Laplacian matrix is exploited to examine whether it is r-structurally balanced or unbalanced. Given the r-structural balance condition, an indicator vector is constructed to calculate the steady states of signed networks. This vector particularly helps to distinguish rooted nodes and non-rooted nodes from all nodes of signed digraphs, determine the connectivity of signed digraphs and develop algebraic criteria for weight balance and sign-average consensus of signed networks. Further, an algorithm via algebraic manipulations is proposed to identify the structural balance and unbalance of signed digraphs, which contributes to achieving behaviour analysis of signed networks. Simulations are included to illustrate validity of the proposed algebraic criteria.

KEYWORDS:

• Signed networks
• algebraic criteria
• structure balance
• behaviour analysis
• antagonistic interactions

Acknowledgments

The authors would like to thank the associate editor and three anonymous reviewers for their insightful comments and suggestions which greatly improved the quality and presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61873013, Grant 61573034, Grant 61922007 and Grant 61520106010, and in part by the Fundamental Research Funds for the Central Universities under Grant YWF-19-BJ-J-42 and Grant YWF-18-BJ-Y-25.

Notes on contributors

Mingjun Du

Mingjun Du received the B.S. degree in Electrical Engineering and Automation from Chengdu University of Information Technology, Chengdu, China, in 2012, and the M.S. degree in Mathematics from Beihang University, Beijing, China, in 2015, where he is currently pursuing the Ph.D. degree with the School of Automation Science and Electrical Engineering, Beihang University. His current research interest includes distributed control of multi-agent systems.

Baoli Ma

Baoli Ma received B.S. and M.S. degrees in Electrical Engineering and Control Engineering from Northwestern Polytechnic University, Xi'an, China. He received Ph.D. degree in System and Control Science from Beihang University, Beijing, China. He is currently a professor with Beihang University. His current research interests include nonlinear control, robotics and automation.

Deyuan Meng

Deyuan Meng received the B.S. degree in Mathematics and Applied Mathematics from Ocean University of China, Qingdao, China, in 2005, and the Ph.D. degree in Control Theory and Control Engineering from Beihang University, Beijing, China, in 2010. He is currently with the Seventh Research Division and School of Automation Science and Electrical Engineering at Beihang University. From November 2012 to November 2013, he was a visiting scholar with the Department of Electrical Engineering and Computer Science, Colorado School of Mines, Golden, CO, USA. His current research interests include iterative learning control, multi-agent systems and social opinion dynamics.