Abstract: We study the impact of the distance between two hubs on network coherence defined by Laplacian eigenvalues. Network coherence is a measure of the extent of consensus in a linear system with additive noise. To obtain an exact determination of coherence based on the distance, we choose a family of tree networks with two hubs controlled by two parameters. Using the tree’s regular structure, we obtain analytical expressions of the coherences with regard to network parameters and the network size. We then demonstrate that a shorter distance and a larger difference in the degrees of the two hubs lead to a higher coherence. With the same network size and distance, the best coherence occurs in the tree with the largest difference in the hub’s degrees. Finally, we establish a correlation between network coherence and average path length and find that they behave linearly.

中心节点距离对树状网络一致性的影响

[1]Andrade JS, Herrmann HJ, Andrade RFS, et al., 2005. Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. Phys Rev Lett, 94(1):018702.

[2]Bamieh B, Jovanovic MR, Mitra P, et al., 2012. Coherence in large-scale networks: dimension-dependent limitations of local feedback. IEEE Trans Autom Contr, 57(9):2235-2249.

[3]Barabási AL, Ravasz E, Vicsek T, 2001. Deterministic scale-free networks. Phys A Stat Mech Appl, 299(3-4):559-564.

[4]Chen J, Sun WG, Wang J, 2023. Topology design for leader-follower coherence in noisy asymmetric networks. Phys Scr, 98(1):015215.

[5]Comellas F, Sampels M, 2002. Deterministic small-world networks. Phys A Stat Mech Appl, 309(1-2):231-235.

[6]Comellas F, Ozón J, Peters JG, 2000. Deterministic small-world communication networks. Inform Process Lett, 76(1-2):83-90.

[7]Dai MF, Wang XQ, Zong Y, et al., 2017. First-order network coherence and eigentime identity on the weighted Cayley networks. Fractals, 25(5):1750049.

[8]Gao HP, Zhu J, Chen X, et al., 2022. Coherence analysis of symmetric star topology networks. Front Phys, 10:876994.

[9]Gao L, Peng JH, Tang CM, 2021. Optimizing the first-passage process on a class of fractal scale-free trees. Fract Fract, 5(4):184.

[10]Gao W, Yan L, Li YF, et al., 2022a. Network performance analysis from binding number prospect. J Amb Intell Human Comput, 13:1259-1267.

[11]Gao W, Chen YJ, Zhang YQ, 2022b. Viewing the network parameters and H-factors from the perspective of geometry. Int J Intell Syst, 37(10):6686-6728.

[12]Grone R, Merris R, Sunder V, 1990. The Laplacian spectrum of a graph. SIAM J Matr Anal Appl, 11(2):218-238.

[13]Hong MD, Sun WG, Liu SY, et al., 2020. Coherence analysis and Laplacian energy of recursive trees with controlled initial states. Front Inform Technol Electron Eng, 21(6):931-938.

[14]Hu TC, Li LL, Wu YQ, et al., 2022. Consensus dynamics in noisy trees with given parameters. Mod Phys Lett B, 36(7):2150608.

[15]Hu X, Zhang ZF, Li CD, 2021. Consensus of multi-agent systems with dynamic join characteristics under impulsive control. Front Inform Technol Electron Eng, 22(1):120-133.

[16]Imran M, Hafi S, Gao W, et al., 2017. On topological properties of Sierpinski networks. Chaos Sol Fract, 98:199-204.

[17]Jing T, Yang L, Sun WG, 2021. Exact calculations of network coherence in weighted ring-trees networks and recursive trees. Phys Scr, 96(8):085217.

[18]Karayannakis D, Aivalis CJ, 2018. Reciprocal Vieta-type formulas and some applications. J Discr Math Sci Cryptogr, 21(1):35-39.

[19]Li QS, Zaman S, Sun WT, et al., 2020. Study on the normalized Laplacian of a penta-graphene with applications. Int J Quant Chem, 120(9):e26154.

[20]Liu JB, Bao Y, Zheng WT, et al., 2021. Network coherence analysis on a family of nested weighted n-polygon networks. Fractals, 29(8):2150260.

[21]Liu JB, Bao Y, Zheng WT, 2022. Analyses of some structural properties on a class of hierarchical scale-free networks. Fractals, 30(7):2250136.

[22]Lu MB, Liu L, 2019. Leader-following consensus of second-order nonlinear multi-agent systems subject to disturbances. Front Inform Technol Electron Eng, 20(1):88-94.

[23]Newman MEJ, 2003. The structure and function of complex networks. SIAM Rev, 45(2):167-256.

[24]Olfati-Saber R, Murray RM, 2004. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Contr, 49(9):1520-1533.

[25]Patterson S, Bamieh B, 2014. Consensus and coherence in fractal networks. IEEE Trans Contr Netw Syst, 1(4):338-348.

[26]Peng JH, Xiong J, Xu GA, 2014. Analysis of diffusion and trapping efficiency for random walks on non-fractal scale-free trees. Phys A Stat Mech Appl, 407:231-244.

[27]Rao S, Ghose D, 2014. Sliding mode control-based autopilots for leaderless consensus of unmanned aerial vehicles. IEEE Trans Contr Syst Technol, 22(5):1964-1972.

[28]Ren W, Beard RW, Atkins EM, 2007. Information consensus in multivehicle cooperative control. IEEE Contr Syst Mag, 27(2):71-82.

[29]Sun WG, Ding QY, Zhang JY, et al., 2014. Coherence in a family of tree networks with an application of Laplacian spectrum. Chaos, 24(4):043112.

[30]Wang L, Liu ZX, 2009. Robust consensus of multi-agent systems with noise. Sci China Ser F Inform Sci, 52(5):824-834.

[31]Xiao L, Boyd S, Kim SJ, 2007. Distributed average consensus with least-mean-square deviation. J Parall Distrib Comput, 67(1):33-46.

[32]Yi YH, Yang BJ, Zhang ZB, et al., 2022. Biharmonic distance-based performance metric for second-order noisy consensus networks. IEEE Trans Inform Theory, 68(2):1220-1236.

[33]Yu WW, Chen GR, Cao M, 2010. Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica, 46(6):1089-1095.

[34]Yu XD, Zaman S, Ullah A, et al., 2023. Matrix analysis of hexagonal model and its applications in global mean-first-passage time of random walks. IEEE Access, 11:10045-10052.

[35]Zaman S, 2022. Spectral analysis of three invariants associated to random walks on rounded networks with 2n-pentagons. Int J Comput Math, 99(3):465-485.

[36]Zaman S, Ullah A, 2023. Kemeny’s constant and global mean first passage time of random walks on octagonal cell network. Math Methods Appl Sci, 46(8):9177-9186.

[37]Zaman S, Koam ANA, Khabyah AA, et al., 2022. The Kemeny’s constant and spanning trees of hexagonal ring network. CMC-Comput Mater Contin, 73:6347-6365.

[38]Zhang HF, Zhang J, Zhou CS, et al., 2010. Hub nodes inhibit the outbreak of epidemic under voluntary vaccination. New J Phys, 12(2):023015.

[39]Zhang LZ, Li YY, Lou JG, et al., 2022. Bipartite asynchronous impulsive tracking consensus for multi-agent systems. Front Inform Technol Electron Eng, 23(10):1522-1532.

[40]Zhang ZZ, Zhou SG, Xie WL, et al., 2009. Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect. Phys Rev E, 79(6):061113.

[41]Zhu J, Huang D, Jiang HJ, et al., 2021. Synchronizability of multi-layer variable coupling windmill-type networks. Mathematics, 9(21):2721.