Figure 2. Reconstruction of two higher order properties of directed weighted networks: the average nearest neighbor strength anns [panels (a,c)] and the weighted clustering coefficient c(w) [panels (b,d)] (refer to³⁷ for their formal definition).

Scatter plots of these quantities observed for the real network G₀ and their ensemble averages obtained from the degree-corrected gravity model of eq. (8) (red circles) or from the standard gravity model of eq. (1) (green asterisks). Upper panels (a,b) refer to WTW, lower panels (c,d) to E-mid. Remarkably, our degree-corrected gravity allows to obtain fairly accurate estimates for the anns, whereas, the standard gravity model completely fails in this respect as the resulting reconstructed network is fully connected. The degree-corrected gravity model outperforms the standard gravity model also in the reconstruction of c(w). In this latter case, note that systematically overestimates the real c(w), because in the definition of this quantity the number of reciprocal links plays an important role, yet it is slightly underestimated by the method (see Figs 5 and 6, and refer to the discussion in²⁶).