1999/05/11 by Cristian F. Moukarzel, Moukarzel, Cristian F., Márcio Argollo de Menezes +2
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Graph theory and applications #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/9905131
4 pages, uses psfig
arxiv created 1999/05/11 · arxiv updated 2009/11/30
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-pathscond-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 ℓ(r) behave as ℓ(r) ~ r for r < rc, and as ℓ(r) ~ rc for r > rc, where r is the Euclidean separation between two points and rc(p,L) = p-1/d log(Ldp) 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 rc(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.