The hierarchical small-world network effectively models the benefit transmission web of pyramid schemes in China and many other countries. In this paper, by applying spectral graph theory, we studied three important aspects of the consensus problem in such networks: convergence speed, communication time-delay robustness, and network coherence. First, we explicitly determined the Laplacian eigenvalues of the network by making use of its tree-like structure. Second, we found that the consensus algorithm on the hierarchical small-world network converges faster than on some well studied sparse networks but is less robust to time delays. We also derived a closed-form for the first-order network coherence. Our results show that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Finally, we argued that some network structure characteristics, such as a large maximum degree, small average path length, and high vertex and edge connectivity, are responsible for the strong robustness against external perturbations.
Citation: Yunhua Liao, Mohamed Maama, M. A. Aziz-Alaoui. Consensus dynamics and coherence in hierarchical small-world networks[J]. Networks and Heterogeneous Media, 2025, 20(2): 482-499. doi: 10.3934/nhm.2025022
| [1] | Chentong Li, Jinyan Wang, Jinhu Xu, Yao Rong . The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments. Mathematical Biosciences and Engineering, 2022, 19(5): 4690-4702. doi: 10.3934/mbe.2022218 |
| [2] | Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva . An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences and Engineering, 2010, 7(1): 123-147. doi: 10.3934/mbe.2010.7.123 |
| [3] | Matthew D. Johnston, Bruce Pell, David A. Rubel . A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity. Mathematical Biosciences and Engineering, 2023, 20(9): 16083-16113. doi: 10.3934/mbe.2023718 |
| [4] | Ali Mai, Guowei Sun, Lin Wang . The impacts of dispersal on the competition outcome of multi-patch competition models. Mathematical Biosciences and Engineering, 2019, 16(4): 2697-2716. doi: 10.3934/mbe.2019134 |
| [5] | Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin . Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068 |
| [6] | Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis . A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme. Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679 |
| [7] | Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283 |
| [8] | Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135 |
| [9] | Yanxia Dang, Zhipeng Qiu, Xuezhi Li . Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences and Engineering, 2017, 14(4): 901-931. doi: 10.3934/mbe.2017048 |
| [10] | Azmy S. Ackleh, Shuhua Hu . Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences and Engineering, 2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133 |
The hierarchical small-world network effectively models the benefit transmission web of pyramid schemes in China and many other countries. In this paper, by applying spectral graph theory, we studied three important aspects of the consensus problem in such networks: convergence speed, communication time-delay robustness, and network coherence. First, we explicitly determined the Laplacian eigenvalues of the network by making use of its tree-like structure. Second, we found that the consensus algorithm on the hierarchical small-world network converges faster than on some well studied sparse networks but is less robust to time delays. We also derived a closed-form for the first-order network coherence. Our results show that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Finally, we argued that some network structure characteristics, such as a large maximum degree, small average path length, and high vertex and edge connectivity, are responsible for the strong robustness against external perturbations.
| [1] | M. Degroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118–121. https://doi.org/10.1080/01621459.1974.10480137 |
| [2] |
X. Xu, Z. Du, X. Chen, C. Cai, Confidence consensus-based model for large-scale group decision making: A novel approach to managing non-cooperative behaviors, Inf. Sci., 477 (2019), 410–427. https://doi.org/10.1016/j.ins.2018.10.058 doi: 10.1016/j.ins.2018.10.058
|
| [3] |
Q. Zha, X. He, M. Zhan, N. Lang, Managing consensus in balanced networks based on opinion and trust/distrust evolutions, Inf. Sci., 643 (2023), 119223. https://doi.org/10.1016/j.ins.2023.119223 doi: 10.1016/j.ins.2023.119223
|
| [4] |
R. Olfati-Saber, R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520–1533. https://doi.org/10.1109/TAC.2004.834113 doi: 10.1109/TAC.2004.834113
|
| [5] |
S. Patterson, B. Bamieh, Consensus and coherence in fractal networks, IEEE Trans. Control Network Syst., 1 (2014), 338–348. https://doi.org/10.1109/TCNS.2014.2357552 doi: 10.1109/TCNS.2014.2357552
|
| [6] |
Y. Shang, A system model of three-body interactions in complex networks: Consensus and conservation, Proc. R. Soc. A, 478 (2022), 20210564. http://doi.org/10.1098/rspa.2021.0564 doi: 10.1098/rspa.2021.0564
|
| [7] |
F. Meng, J. Tang, P. Wang, X. Chen, A programming-based algorithm for interval-valued intuitionistic fuzzy group decision making, Knowl.-Based Syst., 144 (2018), 122–143. https://doi.org/10.1016/j.knosys.2017.12.033 doi: 10.1016/j.knosys.2017.12.033
|
| [8] |
X. Zhou, W. Liang, S. Huang, M. Fu, Social recommendation with large-scale group decision-making for cyber-enabled online service, IEEE Trans. Comput. Social Syst., 6 (2019), 1073–1082. https://doi.org/10.1109/TCSS.2019.2932288 doi: 10.1109/TCSS.2019.2932288
|
| [9] |
C. Zhang, Z. Ni, Y. Xu, E. Luo, L. Chen, Y. Zhang, A trustworthy industrial data management scheme based on redactable blockchain, J. Parallel Distrib. Comput., 152 (2021), 167–176. https://doi.org/10.1016/j.jpdc.2021.02.026 doi: 10.1016/j.jpdc.2021.02.026
|
| [10] |
H. Chen, G. Tan, Adaptive iteration localization algorithm based on RSSI in wireless sensor networks, Cluster Comput., 22 (2019), 3059–3067. https://doi.org/10.1007/s10586-018-1875-y doi: 10.1007/s10586-018-1875-y
|
| [11] |
P. Zhang, J. Wang, K. Guo, Compressive sensing and random walk based data collection in wireless sensor networks, Comput. Commun., 129 (2018), 43–53. https://doi.org/10.1016/j.comcom.2018.07.026 doi: 10.1016/j.comcom.2018.07.026
|
| [12] |
D. Sumpter, J. Krause, R. James, I. Couzin, S. Ward, Consensus decision making by fish, Curr. Biol., 18 (2008), 1773–1777. https://doi.org/10.1016/j.cub.2008.09.064 doi: 10.1016/j.cub.2008.09.064
|
| [13] |
B. Nabet, N. Leonard, I. Couzin, S. Levin, Dynamics of decision making in animal group motion, J. Nonlinear. Sci., 19 (2009), 399–435. https://doi.org/10.1007/s00332-008-9038-6 doi: 10.1007/s00332-008-9038-6
|
| [14] |
L. Giraldo, K. Passino, Dynamic task performance, cohesion, and communications in human groups, IEEE Trans. Cybern., 46 (2016), 2207–2219. https://doi.org/10.1109/TCYB.2015.2470225 doi: 10.1109/TCYB.2015.2470225
|
| [15] |
W. Sun, Y. Li, S. Liu, Noisy consensus dynamics in windmill-type graphs, Chaos, 30 (2020), 123131. https://doi.org/10.1063/5.0020696 doi: 10.1063/5.0020696
|
| [16] |
X. Wang, H. Xu, C. Ma, Consensus problems in weighted hierarchical graphs, Fractals, 27 (2019), 1950086. https://doi.org/10.1142/S0218348X19500865 doi: 10.1142/S0218348X19500865
|
| [17] |
W. Sun, M. Hong, S. Liu, K. Fan, Leader-follower coherence in noisy ring-trees networks, Nonlinear Dyn., 102 (2020), 1657–1665. https://doi.org/10.1007/s11071-020-06011-9 doi: 10.1007/s11071-020-06011-9
|
| [18] |
Y. Yang, G. Li, D. Li, J. Zhang, P. Hu, L. Hu, Integrating fuzzy clustering and graph convolution network to accurately identify clusters from attributed graph, IEEE Trans. Network Sci. Eng., 12 (2025), 1112–1125. https://doi.org/10.1109/TNSE.2024.3524077 doi: 10.1109/TNSE.2024.3524077
|
| [19] |
G. Li, B. Zhao, X. Su, Y. Yang, P. Hu, X. Zhou, et al., Discovering consensus regions for interpretable identification of RNA N6-Methyladenosine modification sites via graph contrastive clustering, IEEE J. Biomed. Health Inf., 28 (2024), 2362–2372. https://doi.org/10.1109/JBHI.2024.3357979 doi: 10.1109/JBHI.2024.3357979
|
| [20] |
Y. Yang, X. Su, B. Zhao, G. D. Li, P. Hu, J. Zhang, et al., Fuzzy-based deep attributed graph clustering, IEEE Trans. Fuzzy Syst., 32 (2024), 1951–1964. https://doi.org/10.1109/TFUZZ.2023.3338565 doi: 10.1109/TFUZZ.2023.3338565
|
| [21] |
Y. Shang, On the delayed scaled consensus problems, Appl. Sci., 7 (2017), 713. https://doi.org/10.3390/app7070713 doi: 10.3390/app7070713
|
| [22] |
Y. Qi, Z. Zhang, Y. Yi, H. Li, Consensus in self-similar hierarchical graphs and sierpinski graphs: Convergence speed, delay robustness, and coherence, IEEE Trans. Cybern., 49 (2019), 592–603. https://doi.org/10.1109/TCYB.2017.2781714 doi: 10.1109/TCYB.2017.2781714
|
| [23] |
X. Li, J. Cai, J. Yang, L. Guo, S. Huang, Y. Yi, Performance analysis of delay distribution and packet loss ratio for body-to-body networks, IEEE Internet Things J., 8 (2021), 16598–16612. https://doi.org/10.1109/JIOT.2021.3075578 doi: 10.1109/JIOT.2021.3075578
|
| [24] |
J. Wang, C. Hu, A. Liu, Comprehensive optimization of energy consumption and delay performance for green communication in internet of things, Mob. Inf. Syst., 2017 (2017), 3206160. https://doi.org/10.1155/2017/3206160 doi: 10.1155/2017/3206160
|
| [25] |
B. Bamieh, M. Jovanović, P. Mitra, S. Patterson, Coherence in largescale networks: Dimension dependent limitations of local feedback, IEEE Trans. Autom. Control, 57 (2012), 2235–2249. https://doi.org/10.1109/TAC.2012.2202052 doi: 10.1109/TAC.2012.2202052
|
| [26] | C. Godsil, G. Royle, Algebraic Graph Theory, Springer Science & Business Media, New York, 2013. https://doi.org/10.1007/978-1-4613-0163-9 |
| [27] | R. Bapat, Graph and Matrices, Springer, New York, 2010. https://doi.org/10.1007/978-1-4471-6569-9 |
| [28] |
D. Zelazo, S. Schuler, F. Allgower, Performance and design of cycles in consensus networks, Syst. Control Lett., 62 (2013), 85–96. https://doi.org/10.1016/j.sysconle.2012.10.014 doi: 10.1016/j.sysconle.2012.10.014
|
| [29] |
Z. Wu, Z. Guan, Time-delay robustness of consensus problems in regular and complex networks, Int. J. Mod. Phys. C, 18 (2007), 1339–1350. https://doi.org/10.1142/S0129183107011364 doi: 10.1142/S0129183107011364
|
| [30] | T. Summers, I. Shames, J. Lygeros, F. Dorfler, Topology design for optimal network coherence, in 2015 European Control Conference (ECC), (2015), 575–580. https://doi.org/10.1109/ECC.2015.7330605 |
| [31] | G. Young, L. Scardovi, N. Leonard, Robustness of noisy consensus dynamics with directed communication, in Proceedings of the 2010 American Control Conference, (2010), 6312–6317. https://doi.org/10.1109/ACC.2010.5531506 |
| [32] |
Z. Zhang, B. Wu, Spectral analysis and consensus problems for a class of fractal network models, Phys. Scr., 95 (2020), 085210. https://doi.org/10.1088/1402-4896/ab9e96 doi: 10.1088/1402-4896/ab9e96
|
| [33] |
Y. Yi, Z. Zhang, Y. Lin, G. Chen, Small-world topology can significantly improve the performance of noisy consensus in a complex network, Comput. J., 58 (2015), 3242–3254. https://doi.org/10.1093/comjnl/bxv014 doi: 10.1093/comjnl/bxv014
|
| [34] |
Q. Ding, W. Sun, F. Chen, Network coherence in the web graphs, Commun. Nonlinear Sci. Numer. Simul., 27 (2015), 228–236. https://doi.org/10.1016/j.cnsns.2015.03.011 doi: 10.1016/j.cnsns.2015.03.011
|
| [35] |
W. Sun, Q. Ding, J. Zhang, F. Chen, Coherence in a family of tree networks with an application of Laplacian spectrum, Chaos, 24 (2014), 043112. https://doi.org/10.1063/1.4897568 doi: 10.1063/1.4897568
|
| [36] |
W. Sun, T. Xuan, S. Qin, Laplacian spectrum of a family of recursive trees and its applications in network coherence, J. Stat. Mech.: Theory Exp., 2016 (2016), 063205. https://doi.org/10.1088/1742-5468/2016/06/063205 doi: 10.1088/1742-5468/2016/06/063205
|
| [37] |
Y. Yi, Z. Zhang, L. Shan, G. Chen, Robustness of first- and second-order consensus algorithms for a noisy scale-free samll-world Koch network, IEEE Trans. Control Syst. Technol., 25 (2017), 342–350. https://doi.org/10.1109/TCST.2016.2550582 doi: 10.1109/TCST.2016.2550582
|
| [38] |
M. Dai, J. He, Y. Zong, T. Ju, Y. Sun, W. Su, Coherence analysis of a class of weighted networks, Chaos, 28 (2018), 043110. https://doi.org/10.1063/1.4997059 doi: 10.1063/1.4997059
|
| [39] |
Y. Yi, Z. Zhang, S. Patterson, Scale-free loopy structure is resistant to noise in consensus dynamics in complex networks, IEEE Trans. Cybern., 50 (2020), 190–200. https://doi.org/10.1109/TCYB.2018.2868124 doi: 10.1109/TCYB.2018.2868124
|
| [40] | S. Dorogovtsev, A. Goltsev, J. Mendes, Pseudofractal scale-free web, Phys. Rev. E, 65 (2002), 066122. https://doi.org/10.1103/PhysRevE.65.066122 |
| [41] |
Z. Zhang, F. Comellas, Farey graphs as models for complex networks, Theor. Comput. Sci., 412 (2011), 865–875. https://doi.org/10.1016/j.tcs.2010.11.036 doi: 10.1016/j.tcs.2010.11.036
|
| [42] |
Z. Zhang, S. Gao, L. Chen, S. Zhou, H. Zhang, J. Guan, Mapping Koch curves into scale-free small-world networks, J. Phys. A: Math. Theor., 43 (2010), 395101. https://doi.org/10.1088/1751-8113/43/39/395101 doi: 10.1088/1751-8113/43/39/395101
|
| [43] |
P. Laurienti, K. Joyce, Q. Telesford, J. Burdette, S. Hayasaka, Universal fractal scaling of self-organized networks, Phys. A, 390 (2011), 3608–3613. https://doi.org/10.1016/j.physa.2011.05.011 doi: 10.1016/j.physa.2011.05.011
|
| [44] | M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J., 23 (1973), 298–305. |
| [45] |
R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl., 197 (1994), 143–176. https://doi.org/10.1016/0024-3795(94)90486-3 doi: 10.1016/0024-3795(94)90486-3
|
| [46] |
M. Chen, B. Yu, P. Xu, J. Chen, A new deterministic complex network model with hierarchical structure, Phys. A, 385 (2007), 707–717. https://doi.org/10.1016/j.physa.2007.07.032 doi: 10.1016/j.physa.2007.07.032
|
| [47] |
Y. Shang, Mean commute time for random walks on hierarchical scale-free networks, Internet Math., 8 (2012), 321–337. https://doi.org/10.1080/15427951.2012.685685 doi: 10.1080/15427951.2012.685685
|
| [48] |
R. Brent, H. Kung, On the area of binary tree layouts, Inf. Process. Lett., 11 (1980), 46–48. https://doi.org/10.1016/0020-0190(80)90034-4 doi: 10.1016/0020-0190(80)90034-4
|
| [49] |
G. Zhu, C. Cai, B. Pan, P. Wang, A multi-agent linguistics-style large group decision-making method considering public expectations, Int. J. Comput. Intell. Syst., 14 (2021), 1–13. https://doi.org/10.1007/s44196-021-00037-6 doi: 10.1007/s44196-021-00037-6
|
| [50] |
H. Fu, C. Shiue, The optimal pebbling number of the complete -ary tree, Discrete Math., 222 (2000), 89–100. https://doi.org/10.1016/S0012-365X(00)00008-X doi: 10.1016/S0012-365X(00)00008-X
|
| [51] |
N. Papageorgiou, V. Radulescu, W. Zhang, Multiple solutions with sign information for double-phase problems with unbalanced growth, Bull. London Math. Soc., 57 (2025), 638–656. https://doi.org/10.1112/blms.13218 doi: 10.1112/blms.13218
|
| [52] |
Q. Li, J. Nie, W. Zhang, Multiple normalized solutions for fractional Schrödinger equations with lack of compactness, J. Geom. Anal., 35 (2025), 59. https://doi.org/10.1007/s12220-024-01897-y doi: 10.1007/s12220-024-01897-y
|
| [53] |
A. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512. https://doi.org/10.1126/science.286.5439.509 doi: 10.1126/science.286.5439.509
|
| [54] |
P. Feng, X. Lu, Z. Gong, D. Sun, A case study of the pyramid scheme in China based on communication network, Physica A, 565 (2021), 125548. https://doi.org/10.1016/j.physa.2020.125548 doi: 10.1016/j.physa.2020.125548
|
| [55] |
J. Palacios, J. Renom, Broder and Karlin's formula for hitting times and the Kirchhoff index, Int. J. Quantum Chem., 111 (2011), 35–39. https://doi.org/10.1002/qua.22396 doi: 10.1002/qua.22396
|
| 1. | Yixiang Wu, Necibe Tuncer, Maia Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, 2017, 22, 1553-524X, 1167, 10.3934/dcdsb.2017057 | |
| 2. | Junping Shi, Yixiang Wu, Xingfu Zou, Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction–Diffusion Chemostat Model, 2020, 32, 1040-7294, 1085, 10.1007/s10884-019-09763-0 | |
| 3. | Yixiang Wu, Xingfu Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, 2016, 261, 00220396, 4424, 10.1016/j.jde.2016.06.028 | |
| 4. | Lin Zhao, Zhi-Cheng Wang, Shigui Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, 2020, 51, 14681218, 102966, 10.1016/j.nonrwa.2019.102966 | |
| 5. | Jing Ge, Ling Lin, Lai Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, 2017, 22, 1553-524X, 2763, 10.3934/dcdsb.2017134 | |
| 6. | Yuan Lou, Rachidi B. Salako, Control Strategies for a Multi-strain Epidemic Model, 2022, 84, 0092-8240, 10.1007/s11538-021-00957-6 | |
| 7. | Jinsheng Guo, Shuang-Ming Wang, Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay, 2022, 7, 2473-6988, 6331, 10.3934/math.2022352 | |
| 8. | Rachidi B. Salako, Impact of population size and movement on the persistence of a two-strain infectious disease, 2023, 86, 0303-6812, 10.1007/s00285-022-01842-z | |
| 9. | Yuan Lou, Rachidi B. Salako, Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases, 2023, 370, 00220396, 424, 10.1016/j.jde.2023.06.018 | |
| 10. | Jonas T. Doumatè, Tahir B. Issa, Rachidi B. Salako, Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023213 | |
| 11. | Azmy S. Ackleh, Nicolas Saintier, Aijun Zhang, A multiple-strain pathogen model with diffusion on the space of Radon measures, 2025, 140, 10075704, 108402, 10.1016/j.cnsns.2024.108402 | |
| 12. | Jamal Adetola, Keoni G. Castellano, Rachidi B. Salako, Dynamics of classical solutions of a multi-strain diffusive epidemic model with mass-action transmission mechanism, 2025, 90, 0303-6812, 10.1007/s00285-024-02167-9 |
Yunhua Liao, Mohamed Maama, M. A. Aziz-Alaoui. Consensus dynamics and coherence in hierarchical small-world networks[J]. Networks and Heterogeneous Media, 2025, 20(2): 482-499. doi: 10.3934/nhm.2025022
DownLoad: