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Abstract: We study the consensus of a family of recursive trees with novel features that include the initial states controlled by a parameter. The consensus problem in a linear system with additive noises is characterized as network coherence, which is defined by a Laplacian spectrum. Based on the structures of our recursive treelike model, we obtain the recursive relationships for Laplacian eigenvalues in two successive steps and further derive the exact solutions of first- and second-order coherences, which are calculated by the sum and square sum of the reciprocal of all nonzero Laplacian eigenvalues. For a large network size N, the scalings of the first- and second-order coherences are lnN and N$, respectively. The smaller the number of initial nodes, the better the consensus bears. Finally, we numerically investigate the relationship between network coherence and laplacian energy, showing that the first- and second-order coherences increase with the increase of laplacian energy at approximately exponential and linear rates, respectively.

具有受控初始状态递归树的一致性分析及其拉普拉斯能量

[1]Albert R, Barabási AL, 2002. Statistical mechanics of complex networks. Rev Mod Phys, 74(1):47-97.

[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]Chu ZQ, Liu JB, Li XX, 2016. The Laplacian-energy-like invariants of three types of lattices. J Anal Methods Chem, 2016:7320107.

[4]Dai MF, He JJ, Zong Y, et al., 2018. Coherence analysis of a class of weighted networks. Chaos, 28(4):043110.

[5]Dorogovtsev SN, Mendes JFF, 2002. Evolution of networks. Adv Phys, 51(4):1079-1187.

[6]Dorogovtsev SN, Goltsev AV, Mendes JFF, et al., 2003. Spectra of complex networks. Phys Rev E, 68(4):046109.

[7]Farkas IJ, Derényi I, Barabási AL, et al., 2001. Spectra of “real-world” graphs: beyond the semicircle law. Phys Rev E, 64(2):026704.

[8]Goh KI, Kahng B, Kim D, 2001. Spectra and eigenvectors of scale-free networks. Phys Rev E, 64(5):051903.

[9]Gutman I, Zhou B, 2006. Laplacian energy of a graph. Linear Algebra Appl, 414(1):29-37.

[10]Ma CQ, Li T, Zhang JF, 2010. Consensus control for leader-following multi-agent systems with measurement noises. J Syst Sci Complex, 23(1):35-49.

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

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

[13]Robbiano M, Jimenez R, 2009. Applications of a theorem by Ky fan in the theory of Laplacian energy of graphs. Match-Commun Math Comput Chem, 62(3):537-552.

[14]Russo G, Shorten R, 2018. On common noise-induced synchronization in complex networks with state-dependent noise diffusion processes. Phys D, 369:47-54.

[15]Song L, Huang D, Nguang SK, et al., 2016. Mean square consensus of multi-agent systems with multiplicative noises and time delays under directed fixed topologies. Int J Contr Autom Syst, 14(1):69-77.

[16]Strogatz SH, 2001. Exploring complex networks. Nature, 410(6825):268-276.

[17]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.

[18]Wang LS, Zhang JB, Sun WG, 2018. Adaptive outer synchronization and topology identification between two complex dynamical networks with time-varying delay and disturbance. IMA J Math Contr Inform, in press.

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

[20]Yi YH, Zhang ZZ, Lin Y, et al., 2015. Small-world topology can significantly improve the performance of noisy consensus in a complex network. Comput J, 58(12):3242-3254.

[21]Zhang ZZ, Qi Y, Zhou SG, et al., 2009. Exact solution for mean first-passage time on a pseudofractal scale-free web. Phys Rev E, 79(2):021127.

[22]Zong Y, Dai MF, Wang XQ, et al., 2018. Network coherence and eigentime identity on a family of weighted fractal networks. Chaos Sol Fract, 109:184-194.