Spectral properties of the Laplacian of multiplex networks (1307.2090v1)

Abstract: One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a paper, Gomez et al. [Phys. Rev. Lett. 101, 028701 (2013)] proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks, and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.

Spectral Properties of the Laplacian of Multiplex Networks

The paper of complex networks has been enriched by examining structures known as multiplex networks, where a set of nodes is interconnected in multiple layers, each representing distinct types of interactions or relationships. The paper "Spectral properties of the Laplacian of multiplex networks" examines the role of multiplex topologies in understanding dynamical processes by extending the framework for diffusion processes to take into account various configurations across the network layers. This research investigates the spectral characteristics of the Laplacian matrix of these networks, offering insight into the behavior of dynamic processes like diffusion and synchronization.

Structural Decomposition of the Supra-Laplacian

The authors propose a structural decomposition of the multiplex network's Laplacian, termed the supra-Laplacian, into intralayer and interlayer contributions. This decomposition uses the mathematical properties of the Kronecker product to separate the spectral behavior of the multiplex network. Such a division is crucial to understanding how independent layers and their interconnections affect a multiplex network's overall dynamics.

Spectral Analysis and Perturbation Theory

The spectral analysis of the supra-Laplacian reveals notable characteristics about the network's dynamic processes. By employing perturbation theory, the authors show how the spectrum's eigenvalues can be analyzed under conditions of weak and strong interlayer couplings. They discover that in cases of weak interlayer interactions, eigenvalues corresponding to intralayer structures dominate, with perturbations introduced proportionally by interlayer connections. Conversely, in strong interlayer interactions, these influences reverse, with interlayer properties holding more sway over the spectrum.

Implications for Diffusion Processes and Synchronizability

For diffusion processes, the multiplex structure can lead to faster convergence times compared to any intralayer network alone, indicating a super-diffusive behavior. The paper finds that diffusion time scales are significantly influenced by the multiplex configuration, highlighting how multiplex connectivity can expedite the spread of information or substances across a network.

When applied to synchronization problems in multiplex networks, the authors ascertain that an optimal coupling strength exists in the interlayer connections where synchronization stability is maximized. This finding is crucial for designing networks that require synchronized operations across different functional units, such as in distributed computational systems or multi-agent robotic systems.

Broader Implications and Future Perspectives

The spectral insights gained from this research offer practical consequences for fields requiring robust network connectivity modeling, such as neuroscience, transportation systems, and social networks. By understanding how multiplex structures influence network dynamics, improved strategies can be devised for optimizing processes like information dissemination and cooperative synchronization.

Future developments will likely explore more complex scenarios beyond uniform weight assumptions between layers and delve into dynamic multiplex networks where interlayer and intralayer connections evolve over time. Additionally, considering directed or weighted networks could provide further refinement in understanding multiplex networks' full potential in real-world applications.