Exact analytical results on synchronizability in multilayer networks

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Abstract

Synchronization in multilayer networks plays a fundamental role in many real-world systems; yet, understanding how interlayer structures influence synchronizability remains a major challenge. In this paper, we investigated the synchronizability of multilayer networks with undirected and directed interlayer couplings. By deriving the exact spectral properties of the corresponding supra-Laplacian matrices, explicit synchronizability criteria were established for bounded and unbounded synchronization regions. The results revealed that interlayer coupling patterns strongly affected synchronization behavior. In undirected multilayer networks, increasing the number of layers weakened synchronizability, whereas stronger intra- and interlayer coupling strengths could enhance synchronization performance. In contrast, multilayer networks with directed interlayer couplings exhibited different behavior: Synchronizability became independent of the number of layers, demonstrating the significant impact of coupling asymmetry on collective dynamics. Moreover, simulations on Erdös–Rényi multilayer networks validated the theoretical predictions. These findings provide analytical insights into how multilayer architectures and coupling asymmetry shape synchronization dynamics and may offer guidance for the design and optimization of complex interconnected systems.

Keywords

multilayer networks / synchronization / synchronizability / Laplacian / supra-Laplacian matrix

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References

[1] D. Watts, S. Strogatz, Collective dynamics of small-world networks, Nature, 393(1998), 440-442. https://doi.org/10.1038/30918

[2] A. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286(1999), 509-512. https://doi.org/10.1126/science.286.5439.509

[3] J. Delgado, Emergence of social conventions in complex networks, Artif. Intell., 141(2002), 171-185. https://doi.org/10.1016/S0004-3702(02)00262-X

[4] E. Bullmore, O. Sporns, The economy of brain network organization, Nat. Rev. Neurosci., 13(2012), 336-349. https://doi.org/10.1038/nrn3214

[5] G. A. Pagani, M. Aiello, The power grid as a complex network: A survey, Phys. A, 392(2013), 2688-2700. https://doi.org/10.1016/j.physa.2013.01.023

[6] J. X. Chen, J. Q. Hu, R. Kapral, Chemical logic gates on active colloids, Adv. Sci., 11(2024), 2305695. https://doi.org/10.1002/advs.202305695

[7] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks, Phys. Rep., 469(2008), 93-153. https://doi.org/10.1016/j.physrep.2008.09.002

[8] Z. Wang, Y. Wang, Attack-resilient pulse-coupled synchronization, IEEE Trans. Control Netw. Syst., 6(2019), 338-351. https://doi.org/10.1109/TCNS.2018.2817922

[9] C. Chen, K. Xie, F. L. Lewis, S. Xie, R. Fierro, Adaptive synchronization of multi-agent systems with resilience to communication link faults, Automatica, 111(2020), 108636. https://doi.org/10.1016/j.automatica.2019.108636

[10] J. Gómez-Gardenes, Y. Moreno, A. Arenas, Paths to synchronization on complex networks, Phys. Rev. Lett., 98(2007), 034101. https://doi.org/10.1103/PhysRevLett.98.034101

[11] K. Sone, Y. Ashida, T. Sagawa, Topological synchronization of coupled nonlinear oscillators, Phys. Rev. Res., 4(2022), 023211. https://doi.org/10.1103/PhysRevResearch.4.023211

[12] G. R. Chen, Z. S. Duan, Network synchronizability analysis: A graph-theoretic approach, Chaos, 18(2008), 037102. https://doi.org/10.1063/1.2965530

[13] F. Della Rossa, P. De Lellis, Synchronization and pinning control of stochastic coevolving networks, Annu. Rev. Control, 53(2022), 147-160. https://doi.org/10.1016/j.arcontrol.2022.04.005

[14] R. M. D’Souza, M. Di Bernardo, Y. Y. Liu, Controlling complex networks with complex nodes, Nat. Rev. Phys., 5(2023), 250-262. https://doi.org/10.1038/s42254-023-00566-3

[15] S. Jafarizadeh, D. Veitch, F. Tofigh, J. Lipman, M. Abolhasan, Optimal synchronizability in networks of coupled systems: topological view, IEEE Trans. Netw. Sci. Eng., 8(2021), 1517-1530. https://doi.org/10.1109/TNSE.2021.3063240

[16] J. T. Lizier, F. Bauer, F. M. Atay, J. Jost, Analytic relationship of relative synchronizability to network structure and motifs, Proc. Natl. Acad. Sci., 120(2023), e2303332120. https://doi.org/10.1073/pnas.2303332120

[17] D. Li, W. Sun, Optimizing synchronizability of small-world networks via adjusting edge weights, J. Complex Netw., 13(2025), cnaf031. https://doi.org/10.1093/comnet/cnaf031

[18] J. B. Liu, X. Wang, J. Cao, The coherence and properties analysis of balanced 2^p-ary tree networks, IEEE Trans. Netw. Sci. Eng., 11(2024), 4719-4728. https://doi.org/10.1109/TNSE.2024.3395710

[19] S. Zaman, Spectral analysis of three invariants associated to random walks on rounded networks with 2n-pentagons, Int. J. Comput. Math., 99(2022), 465-485. https://doi.org/10.1080/00207160.2021.1919303

[20] J. Chen, W. Sun, J. Wang, Topology design for leader-follower coherence in noisy asymmetric networks, Phys. Scr., 98(2023), 015215. https://doi.org/10.1088/1402-4896/aca9a3

[21] X. F. Wang, G. R. Chen, Synchronization in small-world dynamical networks, Int. J. Bifurcation Chaos, 12(2002), 187-192. https://doi.org/10.1142/S0218127402004292

[22] M. Barahona, L. M. Pecora, Synchronization in small-world systems, Phys. Rev. Lett., 89(2002), 054101. https://doi.org/10.1103/PhysRevLett.89.054101

[23] X. Zhang, Y. Wu, Synchronizability of multilayer directed dutch windmill networks, Fractal Fract., 6(2022), 537. https://doi.org/10.3390/fractalfract6100537

[24] A. Sole-Ribalta, M. De Domenico, N. E. Kouvaris, A. Diaz-Guilera, S. Gomez, A. Arenas, Spectral properties of the Laplacian of multiplex networks, Phys. Rev. E, 88(2013), 032807. https://doi.org/10.1103/PhysRevE.88.032807

[25] S. Boccaletti, G. Bianconi, R. Criado, C. I. DelGenio, J. Gómez-Garde˜nes, M. Romance, et al., The structure and dynamics of multilayer networks, Phys. Rep., 544(2014), 1-122. https://doi.org/10.1016/j.physrep.2014.07.001

[26] S. Majhi, S. Ghosh, P. K. Pal, S. Pal, T. K. Pal, D. Ghosh, et al., Patterns of neuronal synchrony in higher-order networks, Phys. Life Rev., 52(2025), 144-170. https://doi.org/10.1016/j.plrev.2024.12.013

[27] J. Zhuang, J. Cao, L. Tang, Y. Xia, M. Perc, Synchronization analysis for stochastic delayed multilayer network with additive couplings, IEEE Trans. Syst. Man Cybern. Syst., 50(2020), 4807-4816. https://doi.org/10.1109/TSMC.2018.2866704

[28] P. K. Pal, M. S. Anwar, M. Perc, D. Ghosh, Global synchronization in generalized multilayer higher-order networks, Phys. Rev. Res., 6(2024), 033003. https://doi.org/10.1103/PhysRevResearch.6.033003

[29] X. Wu, X. Wu, C. Y. Wang, B. Mao, J. A. Lu, J. Lü et al., Synchronization in multiplex networks, Phys. Rep., 1060(2024), 1-54. https://doi.org/10.1016/j.physrep.2024.01.005

[30] H. Liu, J. Xu, Z. Li, X. Wang, J. Lü Z. Zeng, Optimizing synchronizability of multilayer networks based on the graph comparison method, IEEE Trans. Circuits Syst. I: Regular Papers, 67(2020), 1740-1751. https://doi.org/10.1109/TCSI.2019.2963687

[31] X. Wei, X. Wu, J. A. Lu, J. Wei, J. Zhao, Y. Wang, Synchronizability of two-layer correlation networks, Chaos, 31(2021), 103124. https://doi.org/10.1063/5.0056482

[32] X. Li, Y. Wu, Synchronizability of multilayer totally homogeneous networks, Mod. Phys. Lett. B, 39(2025), 2450478. https://doi.org/10.1142/S0217984924504785

[33] L. Tang, X. Wu, J. Lü J. A. Lu, R. M. D’Souza, Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators, Phys. Rev. E, 99(2019), 012304. https://doi.org/10.1103/PhysRevE.99.012304

[34] L. Zhang, Y. Wu, Synchronizability of multilayer networks with K-nearest-neighbor topologies, Front. Phys., 8(2020), 571507. https://doi.org/10.3389/fphy.2020.571507

[35] M. M. Xu, J. A. Lu, J. Zhou, Synchronizability and eigenvalues of two-layer star networks, Acta Phys. Sin., 65(2016), 028902. https://doi.org/10.7498/aps.65.028902

[36] J. Li, Y. Luan, X. Wu, J. A. Lu, Synchronizability of double-layer dumbbell networks, Chaos, 31(2021), 073101. https://doi.org/10.1063/5.0049281

[37] J. Hu, Y. Peng, X. Wu, Optimal inter-layer connections for maximizing synchronizability in two-layer chain network, Chaos, 35(2025), 083127. https://doi.org/10.1063/5.0279141

[38] A. Amirkhani, A. H. Barshooi, Consensus in multi-agent systems: A review, Artif. Intell. Rev., 55(2022), 3897-3935. https://doi.org/10.1007/s10462-021-10097-x

[39] Y. Zhang, P. Lin, J. Chen, W. Sun, A hybrid cryptosystem for medical image security: Integrating finite fields with conservative hyperchaos, Nonlinear Dyn., 114(2026), 129. https://doi.org/10.1007/s11071-025-11994-4

[40] R. Bhatia, Matrix Analysis , Springer, New York, 1997. https://doi.org/10.1007/978-1-4612-0653-8