Abstract
Elements composing complex systems usually interact in several different ways, and as such, the interaction architecture is well modeled by a network with multiple layers—a multiplex network—where the system’s complex dynamics is often the result of several intertwined processes taking place at different levels. However, only in a few cases can such multilayered architecture be empirically observed, as one usually only has experimental access to such structure from an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modeled as a single interaction layer or if it results from the aggregation and interplay of multiple layers. Assuming a prior of intralayer Markovian diffusion, here we show that by using local information provided by a random walker navigating the aggregated network, it is possible to determine, in a robust manner, whether these dynamics can be more accurately represented by a single layer or if they are better explained by a (hidden) multiplex structure. In the latter case, we also provide Bayesian methods to estimate the most probable number of hidden layers and the model parameters, thereby fully reconstructing its architecture. The whole methodology enables us to decipher the underlying multiplex architecture of complex systems by exploiting the non-Markovian signatures on the statistics of a single random walk on the aggregated network. In fact, the mathematical formalism presented here extends above and beyond detection of physical layers in networked complex systems, as it 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.
- Received 2 February 2018
- Revised 7 June 2018
DOI:https://doi.org/10.1103/PhysRevX.8.031038
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Network science is a powerful framework for grappling with complex systems—such as biological processes, transportation networks, and the spread of social information—by visualizing the elements of the system as nodes connected by their interactions. Often, the best approach is to use a multiplex network, where types of interactions are separated in different layers. However, it is often challenging to find a method for revealing hidden multiplex networks from empirical observations of the aggregated network. Here, we develop a mathematical and computational theory that predicts such an optimal multiplex model.
Our approach requires only an experimental observation of a random walker navigating the aggregated network, and it capitalizes on the fact that Markovian random walks navigating a hidden multiplex network are effectively non-Markovian when the movement is observed on the projected, aggregated network. This loss of Markovianity suggests the presence of a hidden multiplex structure, which we can subsequently reconstruct.
Our mathematical theory extends beyond random walks in networks and allows for a universal projection of non-Markovian dynamics into effective multiplex models: Each layer in the effective multiplex acts as a different “mode,” and the dynamics is reconstructed by following a random walker that stochastically alternates between modes.
This research provides a solution for the long-standing problem of unfolding hidden multiplex networks. This can be applied to research in mobility of cells, animals, and humans, as well as in other areas such as condensed-matter physics, where the movement of electrons in different materials can be modeled as a random walk in different layers of a multiplex network.
