Hierarchical random networks for optimizing communication complexity of consensus-based networks

doi:10.23919/JSEE.2025.000152

Journal of Systems Engineering and Electronics ›› 2026, Vol. 37 ›› Issue (2): 636-651.doi: 10.23919/JSEE.2025.000152

• CONTROL THEORY AND APPLICATION • Previous Articles

Hierarchical random networks for optimizing communication complexity of consensus-based networks

Yuqi WANG¹^,²(), Yun ZHANG¹^,²(), Yunze CAI¹^,²^,³^,⁴^,*()

¹School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China
²Key Laboratory of System Control and Information Processing, Shanghai 200240, China
³National Key Laboratory of Air-based Information Perception and Fusion, Luoyang 471000, China
⁴State Key Laboratory of Submarine Geoscience, Shanghai 200240, China

Received:2025-05-19 Online:2026-04-18 Published:2026-04-30
Contact: Yunze CAI E-mail:wangyuqi@sjtu.edu.cn;zhang_yun@sjtu.edu.cn;yzcai@sjtu.edu.cn
About author:
WANG Yuqi was born in 1992. He received his B.S. degree from Tianjin University, Tianjin, China in 2014 and M.S. degree from Harbin Engineering University, Harbin, China in 2018. From 2020 to 2021, he was a visiting Ph.D. student at the Department of Information Engineering, University of Florence, Italy. Currently, he is a Ph.D. candidate at Shanghai Jiao Tong University. His research interests include complex network topology, consensus mechanisms, and cooperative tracking in multi-agent systems. E-mail: wangyuqi@sjtu.edu.cn

ZHANG Yun was born in 1996. He received his B.E. degree in automation and M.E. degree in control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2018 and 2021, where he is pursuing his Ph.D. degree. His research interests include differential games, adaptive dynamic programming, and their applications in multi-agent systems. E-mail: zhang_yun@sjtu.edu.cn

CAI Yunze was born in 1975. She received her B.S. and M.S. degrees from Northwestern Polytechnical University, Xi’an, China, in 1997 and 2000, respectively, and Ph.D. degree from Shanghai Jiao Tong University, Shanghai, China, in 2003. She is currently a professor at the School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, and a visiting researcher at the National Key Laboratory of Air-based Information Perception and Fusion. Her research interests are information processing and the application of intelligent decision-making and control in multi-agent systems. E-mail: yzcai@sjtu.edu.cn
Supported by:
This project is support by the Aeronautical Science Foundation of China (20220001057001;20240001057002).

Abstract

Abstract:

The consensus mechanism in multi-agent networks has attracted considerable attention in both control and computer science. However, current advancements in consensus-based control theory lack a general framework to optimize the communication complexity required to reach consensus. This gap highlights the necessity of robust analytical frameworks to advance the field. Our proposed method, termed hierarchical random networks, decomposes the entire network into multiple random sub-swarms and constructs a hierarchical structure among these sub-swarms. First, we establish a simplified condition to ensure the connectivity of hierarchical random networks. Further, we prove that the expected number of network connections in hierarchical random networks can be reduced to its lower bound as the size of sub-swarms approaches the square root of the total number of agents. At the end of the paper, we validate the effectiveness of the proposed network topology through simulation case studies on maneuvering target tracking. The results demonstrate that combining hierarchical random networks with consensus-based filters can achieve maneuvering target tracking while reducing communication complexity.

Key words: multi-agent system, consensus mechanism, random network, communication complexity, hierarchical random network

Yuqi WANG, Yun ZHANG, Yunze CAI. Hierarchical random networks for optimizing communication complexity of consensus-based networks[J]. Journal of Systems Engineering and Electronics, 2026, 37(2): 636-651.

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Method	Conectivity condition	Network architecture	Scalability	Communication complexity
Event-triggered [21]	Unexplored	Peer-to-peer	No	Unpredictable
Hierarchical clustering [22]	Only if $ \bar{\eta}=100\% $	Leader-follower	No	$ \bar{\eta}\ (\|{\cal{N}}\|-1) $
ER random network [11]	Details in Remark 3	Peer-to-peer	Yes	$ \dfrac{1}{2} \bar{\eta}\ \|{\cal{N}}\| \, (\|{\cal{N}}\|-1) $
Multi-class hierarchical [20]	Details in [20] Theorem 2	Peer-to-peer	Yes	Details in [20] Corollary 2
This paper	Details in Corollary 1	Peer-to-peer	Yes	$ \bar{\eta}\ (\|{\cal{N}}\|)^{\tfrac{3}{2}} $

Time/s	0−25	25−50	50−100	100−125	125−150	150−160
$ \ddot{{x}} $	5	5	10$ {\mathrm{cos}}\;0.06\text{π} t $	0	−10	10
$ \ddot{{y}} $	5	−5	−5$ {\mathrm{sin}}\;0.06\text{π} t $	0	0	5

Time/s	160−170	170−180	180−200	200−250	250−300	−
$ \ddot{{x}} $	−5	−10	0	−10cos $ 0.1\text{π} t $	0	−
$ \ddot{{y}} $	5	0	−10	−10$ {\mathrm{sin}}\;0.1\text{π} t $	0	−

Model index	1	2	3	4	5	6	7	8	9
$ \ddot{x} $	0	5	−5	5	−5	10	0	−10	0
$ \ddot{y} $	0	5	5	−5	−5	0	10	0	−10

Metric	DSTSF	HSTSF	−
Average RMSE in case 1	1.278	3.626	−
Max RMSE in case 1	3.839	4.494	−
Average CRMSE in case 1	6.093	34.92	−
Consensus loop times	1	1	−
Communication complexity	2475	495	−

Metric	DCMMF	HCMMF	The 2nd HCMMF
Average RMSE in case 2	1.187	2.935	1.075
Max RMSE in case 2	3.763	3.464	2.112
Average CRMSE in case 2	1.018	15.12	16.04
Consensus loop times	3	7	5
Communication complexity	7425	3465	2475

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Hierarchical random networks for optimizing communication complexity of consensus-based networks

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