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Effect of Perturbation and Topological Structure on Synchronization Dynamics in Multilayer Networks

Rajesh Kumar, Suchi Kumari, Anubhav Mishra

TL;DR

This study addresses how perturbations in intra-layer weights and heterogeneous inter-layer coupling reshape synchronization in weighted multiplex networks. By proposing a trust-based method to compute both intra- and inter-layer edge weights and analyzing the Supra-Laplacian spectrum, the work links topology to the algebraic connectivity $\lambda_2$, synchronization stability $R$, and synchronization time $\tau$ across BA, Power-Law, and CSA real-network multiplexes. Key findings show that intra-layer perturbations drive changes in $\lambda_2$ while inter-layer weights critically influence stability and convergence speed, with topology determining the magnitude and nature of these effects; weighted networks generally exhibit slower synchronization and different optimal coupling than unweighted ones. The results have implications for designing and controlling synchronization in real-world multilayer systems, suggesting that inter-layer weight tuning and awareness of clustering coefficients can optimize diffusion and coherence. Future work could extend to graph energy analyses and broader dynamical processes on weighted multiplex structures.

Abstract

The way the topological structure transforms from a decoupled to a coupled state in multiplex networks has been extensively studied through both analytical and numerical approaches, often utilizing models of artificial networks. These studies typically assume uniform interconnections between layers to simplify the analytical treatment of structural properties in multiplex networks. However, this assumption is not applicable for real networks, where the heterogeneity of link weights is an intrinsic characteristic. Therefore, in this paper, link weights are calculated considering the node's reputation and the impact of the inter-layer link weights are assessed on the overall network's structural characteristics. These characteristics include synchronization time, stability of synchronization, and the second-smallest eigenvalue of the Laplacian matrix (algebraic connectivity). Our findings reveal that the perturbation in link weights (intra-layer) causes a transition in the algebraic connectivity whereas variation in inter-layer link weights has a significant impact on the synchronization stability and synchronization time in the multiplex networks. This analysis is different from the predictions made under the assumption of equal inter-layer link weights.

Effect of Perturbation and Topological Structure on Synchronization Dynamics in Multilayer Networks

TL;DR

This study addresses how perturbations in intra-layer weights and heterogeneous inter-layer coupling reshape synchronization in weighted multiplex networks. By proposing a trust-based method to compute both intra- and inter-layer edge weights and analyzing the Supra-Laplacian spectrum, the work links topology to the algebraic connectivity , synchronization stability , and synchronization time across BA, Power-Law, and CSA real-network multiplexes. Key findings show that intra-layer perturbations drive changes in while inter-layer weights critically influence stability and convergence speed, with topology determining the magnitude and nature of these effects; weighted networks generally exhibit slower synchronization and different optimal coupling than unweighted ones. The results have implications for designing and controlling synchronization in real-world multilayer systems, suggesting that inter-layer weight tuning and awareness of clustering coefficients can optimize diffusion and coherence. Future work could extend to graph energy analyses and broader dynamical processes on weighted multiplex structures.

Abstract

The way the topological structure transforms from a decoupled to a coupled state in multiplex networks has been extensively studied through both analytical and numerical approaches, often utilizing models of artificial networks. These studies typically assume uniform interconnections between layers to simplify the analytical treatment of structural properties in multiplex networks. However, this assumption is not applicable for real networks, where the heterogeneity of link weights is an intrinsic characteristic. Therefore, in this paper, link weights are calculated considering the node's reputation and the impact of the inter-layer link weights are assessed on the overall network's structural characteristics. These characteristics include synchronization time, stability of synchronization, and the second-smallest eigenvalue of the Laplacian matrix (algebraic connectivity). Our findings reveal that the perturbation in link weights (intra-layer) causes a transition in the algebraic connectivity whereas variation in inter-layer link weights has a significant impact on the synchronization stability and synchronization time in the multiplex networks. This analysis is different from the predictions made under the assumption of equal inter-layer link weights.
Paper Structure (20 sections, 19 equations, 14 figures, 3 tables)

This paper contains 20 sections, 19 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Proposed Methodology
  3. Intra-layer and inter-layer Edge Weights Calculation
  4. Computation of intra-layer and inter-layer edge weights:
  5. Stability of Synchronization
  6. Simulation Setup
  7. Result and Analysis
  8. Algebraic Connectivity
  9. Network Layers of Multiplex Network Designed Using BA Model
  10. Network Layers of Multiplex Network Designed Using Power Law Network Model
  11. Multiplex Network Constructed From CS-Aarhus Social Multiplex Network magnani2013 Dataset
  12. Stability of Synchronization
  13. Network Layers of Multiplex Network Designed Using BA Model
  14. Network Layers of Multiplex Network Designed Using Power Law Network Model
  15. Multiplex Network Constructed From CS-Aarhus Social Multiplex Network magnani2013 Dataset
  16. ...and 5 more sections

Figures (14)

  • Figure 1: Schematic representation of multiplex network composed by online social networks de2018fundamentals
  • Figure 2: Plot of Algebraic connectivity $\lambda_2 (d_x,p)$for multiplex network (unweighted) designed using the BA model for the perturbed network layers, with $0.2 \leq p \leq 2.0$ and variation in inter-layer link weights $0.2 \leq d_x \leq 2.0$. Variation in $\lambda_2$ (different color shades) is observed for the given values of $p$ and $d_x$ implying that there is an effect of perturbation in the network layers of the multiplex network.
  • Figure 3: Plot of Algebraic connectivity $\lambda_2 (d_x,p)$for multiplex network (weighted) designed using the BA model for the perturbed network layers, with $0.2 \leq p \leq 2.0$ and variation in inter-layer link weights $0.2 \leq d_x \leq 2.0$. Variation in $\lambda_2$ (different color shades) is observed for the given values of $p$ and $d_x$ implying that there is an effect of perturbation in the network layers of the multiplex network.
  • Figure 4: Plot of Algebraic connectivity $\lambda_2 (d_x,p)$for multiplex network (unweighted) designed using the Power law network model for the perturbed network layers, with $0.2 \leq p \leq 2.0$ and variation in inter-layer link weights $0.2 \leq d_x \leq 2.0$. Variation in $\lambda_2$ (different color shades) is observed for the given values of $p$ and $d_x$ implying that there is an effect of perturbation in the network layers of the multiplex network.
  • Figure 5: Plot of Algebraic connectivity $\lambda_2 (d_x,p)$for multiplex network (weighted) designed using the Power law network model for the perturbed network layers, with $0.2 \leq p \leq 2.0$ and variation in inter-layer link weights $0.2 \leq d_x \leq 2.0$. Variation in $\lambda_2$ (different color shades) is observed for the given values of $p$ and $d_x$ implying that there is an effect of perturbation in the network layers of the multiplex network.
  • ...and 9 more figures