Two-dimensional small-world networks: navigation with local information


Navigation process is studied on a variant of the Watts-Strogatz small world network model embedded on a square lattice. With probability pp, each vertex sends out a long range link, and the probability of the other end of this link falling on a vertex at lattice distance rr away decays as rα r^{-\alpha}. Vertices on the network have knowledge of only their nearest neighbors. In a navigation process, messages are forwarded to a designated target. For α<3\alpha <3 and α2\alpha \neq 2, a scaling relation is found between the average actual path length and pLpL, where LL is the average length of the additional long range links. Given pL>1pL>1, dynamic small world effect is observed, and the behavior of the scaling function at large enough pLpL is obtained. At α=2\alpha =2 and 3, this kind of scaling breaks down, and different functions of the average actual path length are obtained. For α>3\alpha >3, the average actual path length is nearly linear with network size.Comment: Accepted for publication in Phys. Rev.

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