Fig. 1. The trajectories of economies from 1985 to 2017 in the space of the logarithmic relative GDP per capita (g) and the logarithmic relative number of patent applications per capita (n_p). The colored curves and gray curves represent the trajectory of 11 representative economies and the remain economies, respectively. The gray dashed line is the fitting function n_p = 1.12g – 0.82 of all data points. The gray dot dash line roughly distinguishes between the developed economies and the developing economies. Developed economies are mainly in the upper right area, while developing economies are in the lower left. The inset plots the distribution of the deviation Δn_p of each data point from the fitting line, in which the blue dashed line is its Gaussian fitting.

Fig. 2. The change of the regional innovation index I of each economy with the years. The colored curves and gray curves represent 11 representative economies and the remain economies, respectively.

Fig. 3. The regional innovation index I vs. global innovation index (GII) for each economy at 2016. The color of each data point shows the logarithmic relative GDP per capita (g) of each economy.

Fig. 4. The correlations between the average regional innovation index of each country and the average growth rate of relative per capita GDP in the period from 1998 to 2017: (a) \begin{document}$ Z=\left\langle {I} \right\rangle^{\beta} $\end{document}, where β = 3.80 corresponding to the strongest correlation between and Z, and the dashed line is the fitting line. The inset of panel (a) shows the correlation between and (setting β = 1.0); (b) the correlation between and the corrected prediction value Z_f of each economy, where , is the 20-year average of the logarithmic relative GDP per capita g of each economy, and β = 3.80, a = -0.011, b = –0.020, c = 0.0013, and k = 0.033 is the slope of the fitting line in Fig.(a). The dashed line in the inset of Fig. (b) shows the correction function , which is obtained by the fitting for the correlation between ε and , where ε is the regression residuals in the linear regression shown in Fig. (a)

Fig. 5. Designing the moving window length of 1 year, 3 years and 5 years, for given index, the correlation between the average value of the index of each economy and the average growth rate of the relative GDP per capita within the moving window are shown by curves and data points. The black, blue and pink lines and hollow data points show correlation r_I, corresponding to the index I. The different gray dashed lines show the thresholds of the correlation r_I for different level of significance in the case with the minimum data points (corresponding to the case with 1-year moving window length), and the light gray, medium gray and dark gray dashed lines correspond to the significance P = 0.05, 0.01 and 0.001, respectively. The dark yellow, dark cyan, magenta dashed lines and solid data points show correlation r_GII, corresponding to global innovation index (GII). The olive dashed line and hollow data points show correlation r_p, corresponding to the index of the logarithmic relative number of patent applications per capita (n_p) (5-year-moving-window only). The inset shows the correlations between and r_I, and the correlation beween and r_p, where and is the average growth rate of the relative GDP per capita within the moving window for high-income economies and all economies, respectively, and the solid lines respectively are the fitting curve for the data points with the same color.

Fig. 6. (a) and (b) respectively compare the correlations between the average value of the index I of each economy and the average growth rate of the relative GDP per capita at the 5-year moving windows before and after the transition from bottom on r_I wave to peak and the one from peak to bottom. Fig. (a) shows the comparison between 1994 (the cyan hexagons, at the bottom) and 2004 (the pink dots, at the peak), where the data points of the same economy are connected by gray lines, and the green dashed line and the pink dashed line respectively show the linear fittings of 1994 (with a slope of –0.050) and the one of 2004 (with a slope of 0.073); Fig. (b) shows the comparison between 2004 (the pink circles, at the peak) and 2014 (the blue dots, at the valley), where the data points of the same economy are connected by gray lines, and the pink dashed line (with a slope 0.073) and the blue dashed line (with a slope of –0.033) show the linear fittings of 2004 and 2014, respectively; Panel (c) plots the relationship between the change in the valley-peak transition and the change in the peak-valley transition, where the diameter of each circle is proportional to the 20 year average of the economy’s index, and the color corresponds to the 20-year average growth rate of the economy’s relative GDP per capita.