Exact analytical results on synchronizability in multilayer networks

Jing Chen; Dedian Li; Weigang Sun; Xinke Zhao; Jing Chen; Dedian Li; Weigang Sun; Xinke Zhao

doi:10.3934/nhm.2026052

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1376-1392. doi: 10.3934/nhm.2026052

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Research article

Exact analytical results on synchronizability in multilayer networks

1.
School of Information Technology, Zhejiang Yuying College of Vocational Technology, Hangzhou 310018, China
2.
School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China

Received: 24 May 2026 Revised: 20 June 2026 Accepted: 30 June 2026 Published: 03 July 2026

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.
- multilayer networks,
- synchronization,
- synchronizability,
- Laplacian,
- supra-Laplacian matrix
Citation: Jing Chen, Dedian Li, Weigang Sun, Xinke Zhao. Exact analytical results on synchronizability in multilayer networks[J]. Networks and Heterogeneous Media, 2026, 21(4): 1376-1392. doi: 10.3934/nhm.2026052

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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.

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