Super-diffusion in multiplex networks with long-range interactions

This work investigates the emergence of structural super-diffusion in finite multiplex networks, focusing on situations where long-range interactions (LRIs) are present in at least one of the layers. Employing the Lévy random walk model, we explore how the likelihood of observing LRIs and the strength of the coupling among layers, specifically through the inter-layer diffusion coefficient, affect the relaxation time of the entire multiplex network. Our aim is to determine if this collective relaxation time is shorter compared to that in isolated layer configurations. We quantify the relaxation times through algebraic connectivity, the second-smallest eigenvalue of the network’s (Supra-)Laplacian matrix. The formalism is adapted to scenarios where long-range jumps may exist by considering Mellin or Laplace transforms. We study samples of multiplexes whose layers are obtained from the models by Erdös-Rény, Watts-Strogatz and Barabási-Albert, and discuss in great detail the results for multiplexes with two layers. Besides the rather common enhancement of multiplex diffusion in comparison to the diffusion observed in their isolated layers, our results show that the timescale of layers with LRIs may be still faster than that due only to the usual multiplex network, speeding-up the whole system’s diffusion. In addition, we also provide enough evidences that a novel linear diffusion regime emerges with strong inter-layer coupling and the presence of LRIs in one or more layers.

Graphical Abstract highlights the main focus of the work, the occurrence of superdiffusion in a multiplex with \(M=2\) layers. The connectivity ratio \(\eta = \Lambda _2/\textrm{max}(\lambda ^1_2,\lambda ^2_2)\) (red line) crosses the dashed line \(\eta =1\) at a well defined value of \(\mathrm {D_x}\), and remains in the super-diffusion region highlighted in green