From visibility graphs to cognition.

The natural visibility graph method (NVGM) builds complex networks from time series, but earlier work overlooked conditions where the system deviates from ordinary ergodicity due to turbulent events with an inverse power law index μ ranging from 1 to ∞. For 2 < μ < 3, the non-ergodic signal becomes nearly indistinguishable from fractional Gaussian noise (FGN). Using diffusion entropy analysis (DEA) with stripes reveals that meditation significantly reduces the scaling δ. NVGM generates a homogeneous network. The analysis distinguishes sick from healthy patients: in healthy patients, stripes leave δ virtually unchanged, while in sick patients, stripes yield a significantly lower δ, suggesting their heartbeats contain an FGN contribution.

Study at a glance

Abstract

The aim of this study is to discuss the adoption of the popular natural visibility graph method (NVGM) to create complex networks and explore the condition, ignored by early work on this subject, where the dynamical system under analysis is characterized by a deviation from the ordinary ergodic condition, as a consequence of turbulent events (crucial events) with an inverse power law index μ ranging from 1 to ∞ . In the long-time limit for 2 < μ < 3 , the non-ergodic signal becomes virtually indistinguishable from the fractional Gaussian noise (FGN), hypothesized from the NVGM theory. To identify a genuine FGN, the method of statistical analysis, known as diffusion entropy analysis (DEA), with stripes is used. The adoption of DEA with stripes shows that the scaling δ is significantly reduced by meditation. The adoption of NVGM has the effect of generating a homogeneous network. This analysis yields a significant difference between sick and healthy patients. In the case of healthy patients, the adoption of stripes leaves the value of the scaling virtually unchanged, while for sick patients, the adoption of stripes yields a significantly lower value of δ , suggesting that their heartbeats host a FGN contribution. To explain the influence of meditation on δ , it is necessary to make the popular linear response theory by Kubo and his quantum mechanical prescription compatible with ergodicity breaking, using a master equation approach (MEA). The non-Markov MEA hosts ergodicity-breaking crucial events, leading to establish a connection with the literature on macroscopic effects of quantum mechanics. About the paradoxical effect of meditation-induced scaling reduction, it is suggested that more attention should be devoted to the statistical analysis of physiological processes after meditation.