Fig. 4. Robustness of ecosystem service supply and network fragility in empirical networks.

a The relationship between simulated robustness of ecosystem service supply (R_c = R_0.5) and analytically estimated network fragility (f_c = f_0.5) for 251 empirical networks from the Web of Life follows the linear approximation R_c = 1 − f_c. Colour indicates network dispersion [log₁₀(d)]. b The residuals off the linear prediction R_c = 1 − f_c correlate with log₁₀(d), which we model as (R_c − 1 + f_c)/(f_c(1 − f_c)) = ${\lambda }_c$log₁₀(d), where, for the case of R_0.5, ${\lambda }_{0.5}$ = −1.42. c Correcting network fragility for dispersion according to $f_c^{*}=f_c+{\lambda }_c$log₁₀(d) yields an even closer relationship with robustness of ecosystem service supply. d Residuals from R_0.5 − $f_{0.5}^{*}$ by network type show that our model captures well the universal properties, where AF = anemone–fish (n = 10), HP = host–parasite (n = 51), PA = plant–ant (n = 4), PH = plant–herbivore (n = 4), PL = pollination (n = 148), and SD = seed dispersal (n = 34). Error bars show the 5th and 95th quantiles of the data.