Small-world organization as a representation-dependent signature of nonlinear dynamics in biological time series

Small-world topology is usually interpreted in connectivity networks, but graphs derived from individual time series have different node and edge semantics. We examined how three common time-series representations—Quantile Graphs (QG), Gramian Angular Fields (GAF), and Markov Transition Fields (MTF)—shape the topology inferred from single-signal data. We analyzed stochastic and deterministic synthetic series, including a logistic-map benchmark spanning periodic, chaotic, and period-3-window regimes, together with functional magnetic resonance imaging, magnetoencephalography, calcium imaging, simulated microelectrode-array recordings, and physiological signals.

For the primary biological analyses, each representation was converted to a Q-node undirected binary graph at fixed 10% density, and small-worldness was quantified relative to degree-preserving random graphs. Small-world classification was not representation invariant: QG most often yielded \(\sigma >1\), GAF was below the \(\sigma =1\) criterion in most datasets, and MTF showed mixed or near-threshold behavior. In the logistic-map benchmark and an iterative amplitude-adjusted Fourier transform surrogate analysis, topology depended jointly on dynamics and representation.

These results show that small-world organization in graphs derived from individual time series is a representation-dependent signature of nonlinear temporal structure rather than a universal property of biological dynamics. Attention-deficit/hyperactivity disorder Centers of Biomedical Research Excellence Microelectrode array recording simulator Auxiliary lattice/random small-world indices Number of samples in the original time series CT gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project Number 498176770 (light sheet microscope project). ASLOC acknowledges the support of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Grant 2023/06563-9.

Open access funding provided by University of Geneva Department Basic Neuroscience, University of Geneva, Geneva, Switzerland Caroline L. Alves, Camilla Bellone & AmirAli Kalbasi Competence Center Artificial Intelligence, Aschaffenburg University of Applied Sciences, Aschaffenburg, Germany Biomens, Aschaffenburg University of Applied Sciences, Aschaffenburg, Germany Institute of Mathematical and Computer Sciences, University of São Paulo, São Paulo, Brazil Loriz Francisco Sallum & Thaise G. Toutain Physics Institute, Federal University of Bahia, Salvador, Brazil Institute of Biosciences, São Paulo State University, Botucatu, São Paulo, Brazil The authors declare no competing interests.

This study involved the analysis of publicly available, de-identified human datasets and in vitro neuronal culture data. The authors collected no new human participant data. All in vitro procedures complied with applicable institutional and national ethical guidelines.

Publicly available human datasets were used in accordance with the terms of the relevant databases, data-use agreements, and publication policies. During the preparation of this work, the authors used ChatGPT solely for language refinement and readability improvement. After using this tool, the authors carefully reviewed and edited the manuscript as needed and take full responsibility for the content of the published article.