# Multiplex decomposition of non-Markovian dynamics and the hidden layer reconstruction problem

## TL;DR

This paper introduces a method to determine whether complex system interactions are best modeled as a multiplex network or a single layer, using local information from a random walk, and extends this to reconstruct non-Markovian dynamics.

## Contribution

It provides a robust technique to identify hidden multiplex structures and decompose complex dynamics into Markov switching diffusive modes, with proven theorems and real-world applications.

## Key findings

- Successfully reconstructs hidden multiplex network architecture.
- Decomposes non-Markovian dynamics into Markov switching diffusive modes.
- Proves the possibility of exact dynamics reconstruction for finite-order processes.

## Abstract

Elements composing complex systems usually interact in several different ways and as such the interaction architecture is well modelled by a multiplex network. However often this architecture is hidden, as one usually only has experimental access to an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modelled as a single interaction layer or results from the aggregation and interplay of multiple layers. Here we show that using local information provided by a random walker navigating the aggregated network one can decide in a robust way if the underlying structure is a multiplex or not and, in the former case, to determine the most probable number of hidden layers. As a byproduct, we show that the mathematical formalism also provides a principled solution for the optimal decomposition and projection of complex, non-Markovian dynamics into a Markov switching combination of diffusive modes. We validate the proposed methodology with numerical simulations of both (i) random walks navigating hidden multiplex networks (thereby reconstructing the true hidden architecture) and (ii) Markovian and non-Markovian continuous stochastic processes (thereby reconstructing an effective multiplex decomposition where each layer accounts for a different diffusive mode). We also state and prove two existence theorems guaranteeing that an exact reconstruction of the dynamics in terms of these hidden jump-Markov models is always possible for arbitrary finite-order Markovian and fully non-Markovian processes. Finally, we showcase the applicability of the method to experimental recordings from (i) the mobility dynamics of human players in an online multiplayer game and (ii) the dynamics of RNA polymerases at the single-molecule level.