Table 3

Statistics and uncertainty

LocationData structureTest or analysisNUncertainty [CI95%]
Extended Data Fig. 1-1DGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)64,620
64,620
64,620
64,620
λ = 23.8 [23.5, 24.0]
λ = 22.8 [22.7, 22.9]
λ = 22.2 [22.1, 22.2]
λ = 23.4 [23.3, 23.6]
Fig. 2AGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)16,110λ = 23.1 [22.8, 23.3]
Fig. 2BGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)16,110λ = 23.9 [23.7, 24.2]
Fig. 2CGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)32,400λ = 32.8 [32.5, 33.0]
Fig. 2DGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)64,620λ = 23.4 [23.3, 23.6]
Extended Data Fig. 2-2BGaussian predictor
Gaussian response
Linear correlation1065r = −0.14 [−0.20, −0.08]
Fig. 3DGaussian predictor
Exponential response
Nonlinear regression (iterative optimization)12,924λ = 27.8 [27.4, 28.2]
Fig. 3FGaussian predictor
Gaussian response
Linear correlation1065r = 0.70 [0.67, 0.73]
Fig. 4CGaussian predictor
Gaussian response
Linear correlation80r = 0.35 [0.14, 0.53]
Fig. 8AGaussian predictor
Gaussian response
Linear correlation16,110
16,110
32,400
r = −0.10 [−0.12, −0.09]
r = −0.12 [−0.13, −0.10]
r = −0.11 [−0.12, −0.10]
Fig. 8BGaussian predictor
Gaussian response
Linear correlation351
351
66
91
231
231
28
780
780
10
r = −0.17 [−0.27, −0.06]
r = −0.13 [−0.23, −0.02]
r = −0.41 [−0.60, −0.19]
r = −0.26 [−0.44, −0.06]
r = −0.30 [−0.42, −0.18]
r = −0.30 [−0.40, −0.17]
r = −0.56 [−0.77, −0.24]
r = −0.12 [−0.19, −0.05]
r = −0.17 [−0.24, −0.10]
r = −0.74 [−0.93, −0.20]
Fig. 9BGaussian predictor
Gaussian response
Linear correlation19,667
19,667
64,620
r = 0.43 [0.42, 0.44]
r = 0.23 [0.21, 0.24]
r = 0.06 [0.05, 0.07]
Extended Data Fig. 9-1Gaussian predictor
Gaussian response
Linear correlation8483
8483
16,110
8370
8370
16,110
r = 0.42 [0.40, 0.44]
r = 0.22 [0.20, 0.24]
r = 0.06 [0.05, 0.07]
r = 0.40 [0.38, 0.42]
r = 0.22 [0.20, 0.24]
r = 0.11 [0.10, 0.13]
• Where multiple uncertainties are listed for a figure panel, they correspond to the statistics read left-to-right, top-to-bottom in that panel. For Figure 8B, only uncertainties for significant correlations are listed. Uncertainties for Figures 6-8, 10 are not shown. Extended Data Figure 6-1 contains bootstrapped 95% confidence intervals for the 180 means shown in Figure 6, n = 179. Figure 7 shows bootstrapped 95% confidence intervals in gray; the values of these intervals for all distance bins are available in the figure source data at https://doi.org/10.5281/zenodo.4060485. For Figure 10, means across shuffled matrices are only necessary to account for arbitrary ordering among tied edge weights, and the bootstrapped 95% confidence intervals for these means are vanishingly small. The values of these intervals at all network densities are also included in the figure source data. For nonlinear regressions, confidence intervals are estimated using R−1, the inverse R factor from QR decomposition of the Jacobian, the degrees of freedom for error, and the root mean squared error. For linear correlations, the confidence intervals are based on an asymptotic normal distribution of 0.5*log((1+r)/(1–r)), with an approximate variance equal to 1/(N – 3). For descriptive statistics, e.g., means, empirical 95% confidence intervals are estimated by bootstrapping with 2000 iterations.