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تسجيل مستخدم جديدInfinite characteristic length on small-world systems
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It was recently claimed that on d-dimensional small-world networks with a density p of shortcuts, the typical separation s(p) ~ p^{-1/d} between shortcut-ends is a characteristic length for shortest-paths{cond-mat/9904419}. This contradicts an earlier argument suggesting that no finite characteristic length can be defined for bilocal observables on these systems {cont-mat/9903426}. We show analytically, and confirm by numerical simulation, that shortest-path lengths ell(r) behave as ell(r) ~ r for r < r_c, and as ell(r) ~ r_c for r > r_c, where r is the Euclidean separation between two points and r_c(p,L) = p^{-1/d} log(L^dp) is a characteristic length. This shows that the mean separation s between shortcut-ends is not a relevant length-scale for shortest-paths. The true characteristic length r_c(p,L) diverges with system size L no matter the value of p. Therefore no finite characteristic length can be defined for small-world networks in the thermodynamic limit.
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