Non-equilibrium random walks on multiplex networks

TL;DR

This paper introduces a non-equilibrium random walk model on multiplex networks, deriving the governing supra-Markov matrix, analyzing occupation probabilities, and demonstrating that global search can outperform individual layers.

Contribution

The study develops a novel non-equilibrium random walk framework on multiplex networks, including derivation of the supra-Markov matrix and analysis of search efficiency.

Findings

Stationary occupation probability differs from layer averages unless inter-layer transitions are zero.

Approximate equality of occupation probabilities occurs when inter-layer transition probabilities are very small.

Graph mean first passage time can be smaller than that of any individual layer, indicating efficient global search.

Abstract

We introduce a non-equilibrium discrete-time random walk model on multiplex networks, in which at each time step the walker first undergoes a random jump between neighboring nodes in the same layer, and then tries to hop from one node to one of its replicas in another layer. We derive the so-called supra-Markov matrix that governs the evolution of the occupation probability of the walker. The occupation probability at stationarity is different from the weighted average over the counterparts on each layer, unless the transition probabilities between layers vanish. However, they are approximately equal when the transition probabilities between layers are very small, which is given by the first-order degenerate perturbation theory. Moreover, we compute the mean first passage time (MFPT) and the graph MFPT (GrMFPT) that is the average of the MFPT over all pairs of distinct nodes. Interestingly, we find that the GrMFPT can be smaller than that of any layer taken in isolation. The result embodies the advantage of global search on multiplex networks.