Navigation on Power-Law Small World Network with IncompleteInformation

Abstract

We investigate the navigation process on a variant of theWatts–Strogatz small-world network model with local information. Inthe network construction, each vertex of an N×N squarelattice sends out a long-range link with probability p. The otherend of the link falls on a randomly chosen vertex with probabilityproportional to r−α, where r is the lattice distancebetween the two vertices, and α⩾0. The average actualpath length, i.e. the expected number of steps for passing messagesbetween randomly chosen vertex pairs, is found to scale as apower-law function of the network size Nβ, except whenα is close to a specific value αmin, which givesthe highest efficiency of message navigation. For a finite network,the exponent β depends on both α and p, andαmin drops to zero at a critical value of p whichdepends on N. When the network size goes to infinity, βdepends only on α, and αmin is equal to thenetwork dimensionality.