Fig. 6: Edge betweenness centrality determines the network dissipation at local minima in the fluctuating sink model.
From: Discontinuous transition to loop formation in optimal supply networks
a Edge betweenness centrality Np(e) (numbers and colour code), determined with respect to a single source on the left, is closely related to the network dissipation. b The contribution \({\left({N}_{p}(e)\cdot {\sigma }^{2}+{N}_{p}{(e)}^{2}\cdot {\mu }^{2}\right)}^{\frac{\gamma }{\gamma +1}}\) of a single edge to the minimal network dissipation in a tree network \(\langle {D}_{{\rm{tree}}}^{* }\rangle\) as given in Eq. (31) may be used to estimate the actual network dissipation at minima. Parameters used here are given by σ = 0.5, μ = 1 and γ = 0.9. c The tree estimate \(\langle {D}_{{\rm{tree}}}^{* }\rangle\) correlates strongly with the actual network dissipation at local minima 〈D〉 with high-cost γ = 0.7 and low fluctuations σ = 0.5 since on average only 〈NL〉 = 1.44 loops form for this set of parameters (Pearson correlation coefficient of r = 1.0). d Moving to networks containing many loops, 〈NL〉 = 36.66 on average, obtained by minimising the dissipation for lower cost γ = 0.8 and more fluctuations σ = 1.0, the tree estimate still strongly correlates with the dissipation at minima as measured by a Pearson correlation coefficient of r = 0.82. Results were obtained by applying the relaxation method 100 times for each set of parameters where the set of potential edges \({\mathcal{E}}\) forms a triangular network with N = 169 nodes as shown in Fig. 2c.
