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Fig. 3 | Energy Informatics

Fig. 3

From: A comparison of clustering methods for the spatial reduction of renewable electricity optimisation models of Europe

Fig. 3

Consider a symmetric graph \({\mathcal {G}}_0\) with reactances \(x_{(v,w)}\) and without resistances. Above figures show different first iteration choices of the weighted Clauset-Newman-Moore Algorithm into graphs \({\mathcal {G}}_i\), \(i\in \{1,2,3,4\}\) marked in red. Due to the symmetry of \(G_0\), other than the displayed choices for the fist iteration are equivalent. Without clustering, each node in \(G_0\) can be interpreted as a singleton cluster, yielding the initial modularity of \(Q_0\approx -0.1677\). For the four displayed cases, we calculate: \(\begin{array}{c|c} \mathbf {G}_\mathbf{1 }: \mathbf {A_{02} \approx 0.067> \frac{d_0 d_2}{2m}\approx 0.022} &{} G_2: A_{01} = 0.05 > \frac{d_0 d_1}{2m}\approx 0.018\\ \hline G_3: A_{15} = 0< \frac{d_1 d_5}{2m}\approx 0.314 &{} G_4: A_{23} = 0.005 < \frac{d_2 d_3}{2m}\approx 0.372 \end{array}\) Hence, both \(G_1\) and \(G_2\) would improve the modularity, but \(G_1\) is the better choice, as \(A_{02}-\frac{d_0d_2}{2m} > A_{01}-\frac{d_0d_1}{2m}\). \(G_3\) and \(G_4\) are bad choices, reducing modularity and deteriorate the network community. However, if \(x_{(2,3)}\) was much smaller, for example \(x_{(2,3)}=1\), then \(G_4\) would be the best choice for the first iteration

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