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

Abstract 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.