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

A navigation process is studied on a variant of the Watts-Strogatz small-world network model embedded on a square lattice. With probability p, 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 r away decays as 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 alpha < 3 and alpha not equal 2, a scaling relation is found between the average actual path length and pL, where L is the average length of the additional long range links. Given pL > 1, a dynamic small world effect is observed, and the behavior of the scaling function at large enough pL is obtained. At alpha=2 and 3, this kind of scaling breaks down, and different functions of the average actual path length are obtained. For alpha > 3, the average actual path length is nearly linear with network size.