Diffusion Dynamics on Multiplex Networks

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

Figure 1

Example of a multiplex network with $M=2$ layers. Nodes are the same in both layers. The connectivity at each layer is independent of each other, and the connectivity interlayer is from each node to itself (dashed links).

Figure 2

Evolution of the eigenspectra of the toy model presented in Fig. 1 as a function of the coupling $D_x$ (top panel) for $D_1=D_2=1$, and comparison between the second smallest eigenvalues ${\lambda }_2$ of the different Laplacians (bottom panel).