Fig. 3. Phenomenological renormalization-group.

a Variance $V$ of coarse-grained variables as a function of cluster size $K$, average over subjects (black points; error bars indicate SD over subjects, n = 1003). The solid black line indicates least squares power law fit, i.e. $V=K^{\tilde{\alpha }}$. Dashed lines indicate linear ($\tilde{\alpha }=$ 1) and quadratic ($\tilde{\alpha }=$ 2) growths, corresponding to uncorrelated and fully correlated systems, respectively. $⟨\tilde{\alpha }⟩$ indicates the average exponent across subjects. b Silence log-probability, $F=\ln P_{\text{silence}}$, of coarse-grained variables as a function of cluster size, average over subjects (black points; error bars indicate SD over subjects, n = 1003). The solid black line indicates least squares power law fit, i.e. $F={-K}^{\tilde{\beta }}$. The dashed line indicates the prediction for uncorrelated variables ($\tilde{\beta }=$ 1). In (a) and (b), the variance and the silence log-probability were normalized by their corresponding values at coarse-graining step $k=0$ (original system). $\left.⟨\right)\tilde{\beta }\left.⟩\right)$ indicates the average exponent across subjects. c Eigenvalues $\lambda$ of the covariance matrix as a function of their relative rank, for clusters of different sizes, for one example subject. The solid black line indicates least squares power law fit, i.e. $\lambda ={\left(\frac{\text{rank}}{K}\right)}^{-\mu }$, for $\frac{\text{rank}}{K}<0.4$. $⟨\mu ⟩$ indicates the average exponent across subjects. d Estimated exponent $\mu$ for different cluster sizes. Error bars indicate the estimation error of the exponent (for $K>$ 8 error bars are smaller than the symbols). e Distribution of exponents $\tilde{\alpha }$, $\tilde{\beta }$, $\mu$ for single-subject scans (n = 1003). f Least square estimation errors of PRG exponents. g Relative estimation error of exponents; e.g., $\mathrm{\Delta }\tilde{\alpha }/\tilde{\alpha }$, where $\mathrm{\Delta }\tilde{\alpha }$ is the least square estimation error of exponent $\tilde{\alpha }$ (n = 1003). White circles indicate medians. h The power-law fits of $V\left(K\right)$, $F\left(K\right)$, and $\lambda \left(\text{rank}/K\right)$ were compared to those obtained using an exponential function by calculating the ratio between the explained variance of the competing regression models ($R_{EV}$). Ratios >1 favor the power law hypothesis. Violin plots represent the distribution of ratios across subjects (n = 1003). White circles indicate medians.