# Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks

## TL;DR

This paper investigates fractional continuous-time random walks on multiplex networks, revealing an optimal inter-layer coupling that minimizes relaxation time and demonstrating linear mean square displacement growth despite enhanced diffusion conditions.

## Contribution

It provides analytical expressions and numerical analysis of fractional random walks on circulant multiplex networks, highlighting the impact of inter- and intra-layer coefficients on dynamics.

## Key findings

- Relaxation time has a minimum at an optimal inter-layer to intra-layer coefficient ratio.
- Mean square displacement increases linearly with time even under enhanced diffusion.
- Analytical solutions are derived for circulant multiplex networks with finite nodes.

## Abstract

This work analyzes fractional continuous-time random walks on two-layer multiplexes. A node-centric dynamics is used, in which it is assumed a Poisson distribution of a walker to become active, while a jump to one of its neighbors depends on the connection weight. Synthetic multiplexes with well known topology are used to illustrate dynamical features obtained by numerical simulations, while exact analytical expressions are presented for multiplexes assembled by circulant layers with finite number of nodes. Special attention is given to the effect of inter- $D_x$ and intra-layer $D_i$ coefficients on the system's behavior. In opposition to usual discrete time dynamics, the relaxation time has a well defined minimum at an optimal value of $D_x/D_i$. It is found that, even for the enhanced diffusion condition, the walkers mean square displacement increases linearly with time.