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Directionality-induced jamming in multiplex networks

Mateo Bouchet, Alejandro Tejedor, Xiangrong Wang, Yamir Moreno

TL;DR

The paper investigates diffusion on multiplex networks with directed interlayer couplings and shows that interlayer directionality alone can reproduce known diffusion regimes such as $\textbf{superdiffusion}$ and the $\textbf{prime regime}$, while uncovering a new phase called directionality-induced jamming. Using a spectral analysis of the supra-Laplacian $\mathcal{L}$ and perturbation theory in the weak-coupling limit ($D_X \to 0$) and the strong-coupling limit ($D_X \to \infty$), the authors derive how the eigenvalue spectrum reorganizes under both induced and topological directionality, with $\Lambda_2$ controlling the slowest relaxation. The jamming regime arises when asymmetric interlayer coupling causes a degeneracy of the zero eigenvalue, halting convergence to a global steady state and effectively partitioning the network into dynamically disconnected components; this regime can persist at large $D_X$ and can be engineered in real networks via interlayer-weight optimization. The findings highlight interlayer directionality as a fundamental control parameter for diffusion, with implications for the design, regulation, and robustness of interconnected infrastructures such as transportation and communication networks.

Abstract

We study diffusion on multiplex networks with directed interlayer couplings. We demonstrate both numerically and analytically that even with undirected layers, interlayer directionality alone reproduces superdiffusion and the prime regime. We further reveal a new phenomenon, the directionality-induced jamming, whereby directed interlayer links hinder diffusion, fragmenting the system into dynamically disconnected components and preventing convergence to the steady state of the diffusion process. Via an optimization process, we show that this new regime is attainable in both toy models and real-world topologies. These findings underscore the crucial role of interlayer link directionality in shaping the emergent behavior of multiplex systems, with potential implications for the design and control of such systems.

Directionality-induced jamming in multiplex networks

TL;DR

The paper investigates diffusion on multiplex networks with directed interlayer couplings and shows that interlayer directionality alone can reproduce known diffusion regimes such as and the , while uncovering a new phase called directionality-induced jamming. Using a spectral analysis of the supra-Laplacian and perturbation theory in the weak-coupling limit () and the strong-coupling limit (), the authors derive how the eigenvalue spectrum reorganizes under both induced and topological directionality, with controlling the slowest relaxation. The jamming regime arises when asymmetric interlayer coupling causes a degeneracy of the zero eigenvalue, halting convergence to a global steady state and effectively partitioning the network into dynamically disconnected components; this regime can persist at large and can be engineered in real networks via interlayer-weight optimization. The findings highlight interlayer directionality as a fundamental control parameter for diffusion, with implications for the design, regulation, and robustness of interconnected infrastructures such as transportation and communication networks.

Abstract

We study diffusion on multiplex networks with directed interlayer couplings. We demonstrate both numerically and analytically that even with undirected layers, interlayer directionality alone reproduces superdiffusion and the prime regime. We further reveal a new phenomenon, the directionality-induced jamming, whereby directed interlayer links hinder diffusion, fragmenting the system into dynamically disconnected components and preventing convergence to the steady state of the diffusion process. Via an optimization process, we show that this new regime is attainable in both toy models and real-world topologies. These findings underscore the crucial role of interlayer link directionality in shaping the emergent behavior of multiplex systems, with potential implications for the design and control of such systems.
Paper Structure (7 sections, 28 equations, 4 figures)

This paper contains 7 sections, 28 equations, 4 figures.

Table of Contents

  1. Weak coupling ($D_X \to 0$)
  2. Strong coupling ($D_X \to \infty$)
  3. Acknowledgements
  4. Topology of the Toy multiplex network to simulate induced interlayer directionality.
  5. Approximation for Dominant Coupling $D_{12}$ in an induced directionality scenario
  6. Proof of condition over $\lambda_i^{(\pm)}$
  7. Optimization of interlayer link directionality in the London multiplex to induce a jamming regime

Figures (4)

  • Figure 1: Diffusion Dynamics in multiplex networks with induced interlayer directionality. a) Heat map of the smallest non-zero eigenvalue of $\mathcal{L}$, $\Lambda_2$, as a function of the coupling parameters $D_{12}$ and $D_{21}$. b) Evolution of $\Lambda_2$ as function of $D_{21}$ when $D_{12}=100.0$. The network is given in the Supplementary Materials.
  • Figure 2: Diffusion Dynamics in multiplex networks with topological interlayer directionality. Evolution of the eigenvalues of $\mathcal{L}$ versus the coupling parameter $D_X$ for different network configurations: a) and b) are synthetic networks of $N=6$ nodes per layer. c) represents the London transport network Domenico2014, where the nodes are the stations and the links are the connections between them. Layer 1 represents the DLR line, layer 2 the overground, and layer 3 the underground. It is assumed that $D_K=1.0$ for all layers of any configuration.
  • Figure 3: Topology of the multiplex network used for the results presented in Fig. 1 of the main manuscript.
  • Figure 4: Representation of the interlink values. DLR represents the Docklands Light Railway (Layer 1), OGD the overground (Layer 2), and UGD the underground (Layer 3). The size of the arrow represents its value (with a maximum of 1 and a minimum of 0), and its color (along with its direction) represents the direction of the link: blue represents the normal direction (for example, westham UGD $\rightarrow$ DLR in the first case) and red represents the reverse direction (westham DLR $\rightarrow$ UGD in the same example). The numerical values are given in the file "interlinks_values.txt".