Natural connectivity of ER random graphs: (a) vs. p with N = 100 (circles), 500 (triangles) and 1000 (diamonds); (b) vs. N with p = 0.1 (circles), 0.3 (triangles), 0.5 (diamonds). Each symbol corresponds to an average over 1000 realizations. The lines represent the corresponding analytical results according to Eq. (19).
Natural connectivity of regular ring lattices (squares), random regular graphs (triangles) and ER random graphs (diamonds). From bottom to top, the symbols correspond to K = 3, 4, 5, respectively. For both the regular and ER random graphs, each symbol is obtained as an average over 1000 graph realizations. The lines are guides to the eye.
The cross-over size given by Eq. (20) as a function of K, the density of edges.
Natural connectivity during the process of random rewiring (diamonds) and random degree-preserving rewiring (triangles) starting from regular ring lattices with K = 5 for (a) N = 30 (a) and (b) N = 100 (b). The solid lines indicate the values of the natural connectivity for random regular graphs and the dashed lines represent the values for random graphs (Eq. (19)). Each symbol corresponds to an average over 1000 realizations.
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