| 10 | 0 | 49 |
| 下载次数 | 被引频次 | 阅读次数 |
为揭示复杂系统结构与功能的耦合机理，构建一类多层加权边冠网络模型，系统分析其拉普拉斯谱性质及一阶和二阶一致性行为，以期为多智能体系统的鲁棒设计提供理论依据。首先，基于图论与矩阵分析方法，通过克罗内克积运算和分块矩阵特征值分解，推导了多层加权边冠网络拉普拉斯谱。然后，通过拉普拉斯谱进一步得到一阶和二阶一致性精确表达式，并考察层数、权重及因子网络拓扑对一致性行为与鲁棒性的影响机制。最后，作为应用推广，进一步计算了网络的生成树数与拉普拉斯能量。结果表明：二阶一致性较一阶一致性对拓扑变化更为敏感，且增大层数与权重可显著增强网络鲁棒性。对于相同的G2，当因子网络G1具有更高的网络一致性时，多层加权边冠网络■通常具有更高的一阶网络一致性和更低的二阶网络一致性。对于相同的G1，当因子网络G2具有更高的网络一致性时，多层加权边冠网络■通常也具有更高的网络一致性。研究结果阐明了多层加权边冠网络中拓扑结构与一致性之间的内在耦合规律，为通过调节层数与权重灵活调控多智能体系统的协同能力与抗干扰性能提供了理论基础。
Abstract：To reveal the coupling mechanism between structure and function in complex systems， a class of multilayer weighted edge corona network models is constructed， and their Laplacian spectral properties as well as first-order and second-order coherence behaviors are systematically analyzed， aiming to provide a theoretical basis for the robust design of multi-agent systems. Firstly， based on graph theory and matrix analysis， the Laplacian spectrum of the multilayer weighted edge corona network is derived using Kronecker product operations and block matrix eigenvalue decomposition. Then， exact expressions for first-order and second-order coherence are obtained from the Laplacian spectrum， and the influences of the number of layers， weights， and factor network topologies on coherence behavior and robustness are examined. As an application， the number of spanning trees and the Laplacian energy of the network are further computed. The results show that second-order coherence is more sensitive to topological changes than first-order coherence， and increasing the number of layers and weights can significantly enhance network robustness. For the same G2， when the factor network G1 possesses higher network coherence， the multilayer weighted edge corona network ■ generally exhibits higher first-order coherence and lower second-order coherence. For the same G1， when the factor network G2 possesses higher network coherence， the multilayer weighted edge corona network ■ also generally exhibits higher network coherence. These findings clarify the intrinsic coupling law between topology and coherence in multilayer weighted edge corona networks， providing a theoretical foundation for flexibly regulating the cooperative ability and anti-interference performance of multi-agent systems by adjusting the number of layers and weights.
[1] ALBERT R， BARABáSI A L. Statistical mechanics of complex networks[J]. Reviews of Modern Physics， 2002， 74(1): 47-49.
[2] ARTIME O， GRASSIA M， De Domenico M， et al. Robustness and resilience of complex networks[J]. Nature Reviews Physics， 2024， 6(2): 114-131.
[3] HU J X， THOMAS C E， BRUNAK S. Network biology concepts in complex disease comorbidities[J]. Nature Reviews Genetics， 2016， 17(10): 615-629.
[4] HOLOVATCH Y， KENNA R， Thurner S. Complex systems: physics beyond physics[J]. European Journal of Physics， 2017， 38(2): 023002.
[5] TURNER J R， BAKER R M. Complexity theory: an overview with potential applications for the social sciences[J]. Systems， 2019， 7(1): 4.
[6] CVETKOVI? D， SIMI? S K. Towards a spectral theory of graphs based on the signless Laplacian， II[J]. Linear Algebra and Its Applications， 2010， 432(9): 2257-2272.
[7] SHI Z， LIU J. Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum[J]. AIMS. Math.， 2024， 9(3): 5431-5450.
[8] Ye J， Liu M， STANI? Z. Two classes of graphs determined by the signless Laplacian spectrum[J]. Linear Algebra and Its Applications， 2025， 708: 159-172.
[9] BAMIEH B， JOVANOVIC M R， MITRA P， et al. Coherence in large-scale networks: dimension-dependent limitations of local feedback[J]. IEEE Transactions on Automatic Control， 2012， 57(9): 2235-2249.
[10] PATTERSON S， BAMIEH B. Consensus and coherence in fractal networks[J]. IEEE Transactions on Control of Network Systems， 2014， 1(4): 338-348.
[11] JING T， YANG L， SUN W. G. Exact calculations of network coherence in weighted ring-trees networks and recursive trees[J]. Physica Scripta， 2021， 96(8): 085217.
[12] LIU J B， GUAN L， CAO J D， et al. Coherence analysis for a class of polygon networks with the noise disturbance[J]. IEEE Transactions on Systems， Man， and Cybernetics: Systems， 2025， 55(7): 4718-4727.
[13] XU W Y， WU B， ZHANG Z B， et al. Coherence scaling of noisy second-order scale-free consensus networks[J]. IEEE Transactions on Cybernetics， 2022， 52(7): 5923-5934.
[14] YI Y H， ZHANG Z Z， SHAN L R， et al. Robustness of first-and second-order consensus algorithms for a noisy scale-free small-world Koch network[J]. IEEE Transactions on Control Systems Technology， 2017， 25(1): 342-350.
[15] QI Y， ZHANG Z Z， YI Y H， et al. Consensus in self-similar hierarchical graphs and Sierpiński graphs: convergence speed， delay robustness， and coherence[J]. IEEE Transactions on Cybernetics， 2019， 49(2): 592-603.
[16] DAI M F， HE J J， ZONG Y， et al. Coherence analysis of a class of weighted networks[J]. Chaos: An Interdisciplinary Journal of Nonlinear Science， 2018， 28(4): 043110.
[17] CHEN J， SUN W G， WANG J. Topology design for leader-follower coherence in noisy asymmetric networks[J]. Physica Scripta， 2023， 98(1): 015215.
[18] ZHANG Q， SUN W G， XIA Y X. Performance of leader-follower coherence for selecting leaders using node centrality indices[J]. IEEE Transactions on Circuits and Systems II: Express Briefs， 2024， 71(12): 4949-4953.
[19] MACKIN E， PATTERSON S. Optimizing the coherence of a network of networks[J]. IEEE Transactions on Control of Network Systems， 2020， 7(3): 1465-1475.
[20] FRUCHT R， HARARY F. On the corona of two graphs[J]. Aequationes Mathematicae， 1970， 4(3):322-325.
[21] HOU Y， SHIU W C. The spectrum of the edge corona of two graphs[J]. The Electronic Journal of Linear Algebra， 2010， 20(1): 586-594.
[22] LIU X G， ZHOU S M. Spectra of the neighbourhood corona of two graphs[J]. Linear and Multilinear Algebra， 2014， 62(9): 1205-1219.
[23] BARIK S， SAHOO G. On the Laplacian spectra of some variants of corona[J]. Linear Algebra and Its Applications， 2017， 512: 32-47.
[24] VARGHESE R P， SUSHA D. On the spectra of a new duplication based corona of graphs[J]. Notes on Number Theory and Discrete Mathematics， 2021， 27(1): 208-220.
[25] LIU B， LI X， HUANG J J， et al. Controllability of N-duplication corona product networks with Laplacian dynamics[J]. IEEE Transactions on Neural Networks and Learning Systems， 2025， 36(2): 3693-3707.
[26] LIANG W Y， LIAO Y H， LI D Q， et al. Network coherence of N-duplication corona weighted network[J]. Communications in Nonlinear Science and Numerical Simulation， 2026， 152: 109468.
[27] YI Y H， ZHANG Z Z， PATTERSON S. Scale-free loopy structure is resistant to noise in consensus dynamics in complex networks[J]. IEEE Transactions on Cybernetics， 2020， 50(1): 190-200.
[28] ZHOU S Z， SUN Z R， LIU H X. D-index and Q-index for spanning trees with leaf degree at most k in graphs[J]. Discrete Mathematics， 2024， 347(5): 113927.
[29] DAS K C， ALAZEMI A， AN?ELI? M. On energy and Laplacian energy of chain graphs[J]. Discrete Applied Mathematics， 2020， 284: 391-400.
基本信息:
中图分类号:O157.5
引用信息:
[1]刘家保,李启昌.一类多层加权边冠网络的拉普拉斯谱和一致性[J].南通大学学报(自然科学版)().
基金信息:
安徽省自然科学基金项目（2508085MA017）
2026-06-26
2026-06-26
2026-06-26