Revealing Structural Information about Complex Systems from Minimal Data

Much of the world comprises complex dynamical systems, from networks of genes and proteins in biological cells and neural circuits that govern the brain to fiber optic telecommunication networks to continent-spanning power grids. Mathematical models of such systems typically require the number of variables that change with time and jointly characterize the overall respective system, what researchers call “state space dimension.” Often, however, this information is inaccessible because scientists have only experimental access to just some variables – and in extreme cases, just one – from which to infer and extrapolate.

Establishing significant new reconstruction groundwork, Börner et al. demonstrated a robust appro for inferring the state space dimension of a complex dynamical system by measuring only a single variable.

“In principle, our technique can reconstruct arbitrarily high state space dimensions using only data from a single variable,” said author Georg Börner. “In practice, the technique is limited due to unavoidable measurement errors, noise, and inaccuracies in processing the recorded dynamics.”

The researchers used numerical computer simulations for cornerstone mathematical model systems to show that, given appropriate data quality, their method is, in fact, practicable and robust. They also developed some general guidelines for measuring and evaluating system data.

“Although for high-dimensional systems, dimension inference from a single variable may not yet be practical due to the limited precision of recorded data, our work may serve as a starting point for studying dimension inference from a very small number of variables or even just one,” said Börner. “A possible next step would be to study the technique’s applicability to real-world systems with small numbers, such as small biological networks, depending on the quality and quantity of available data.”

Source: “Revealing system dimension from single-variable time series,” by Georg Börner, Hauke Haehne, Jose Casadiego, Marc Timme, Chaos (2023). The article can be accessed at https://doi.org/10.1063/5.0156448 .

This paper is part of the Nonlinear dynamics, synchronization and networks: Dedicated to Juergen Kurths’ 70th birthday Collection, learn more here .