Fig. 4.

Quantifying the spatial effect of edge-density ρ_e on high betweenness nodes a The metric 〈C_θ〉 (Eq. (9)) decreases for denser networks, capturing the tendency of the nodes to be increasingly clustered around their center of mass. b Correspondingly they also become more isotropic around this center as A_θ (Eq. (11)) approaches 1. c The network also becomes increasingly geometric as indicated by the decrease in the average detour factor 〈D〉 (Eq. (12)) measured for the full network, which experiences an abrupt transition around ρ_e∼0.4. The shaded regions represent fluctuations over hundred realizations of the randomization procedure. d The average BC for nodes at a distance r from the barycenter (rescaled to the interval [0,1]), measured in units of r/R where R = 50 is the grid boundary. Curves are colored according to the value of ρ_e. The dashed line corresponds to the analytical calculation for an infinitely dense random geometric graph⁵⁵. The metrics are computed for the networks generated in Fig. 3. (here $⟨\dots ⟩$ indicates averaging over realizations)