Revealing system dimension from single-variable time series

Complex and networked dynamical systems characterize the time evolution of most of the natural and human-made world. The dimension of their state space, i.e., the number of (active) variables in such systems, arguably constitutes their most fundamental property yet is hard to access in general. Recent work [Haehne et al., Phys. Rev. Lett. 122, 158301 (2019)] introduced a method of inferring the state space dimension of a multi-dimensional networked system from repeatedly measuring time series of only some fraction of observed variables, while all other variables are hidden. Here, we show how time series observations of one single variable are mathematically sufficient for dimension inference. We reveal how successful inference in practice depends on numerical constraints of data evaluation and on experimental choices, in particular the sampling intervals and the total duration of observations. We illustrate robust inference for systems of up to N = 10 to N = 100 variables by evaluating time series observations of a single variable. We discuss how the faithfulness of the inference depends on the quality and quantity of collected data and formulate some general rules of thumb on how to approach the measurement of a given system.