Navigation in a small world with local information

Abstract

It is commonly known that there exist short paths between vertices in a network showing the small-world effect. Yet vertices, for example, the individuals living in society, usually are not able to find the shortest paths, due to the very serious limit of information. To study this issue theoretically, here the navigation process of launching messages toward designated targets is investigated on a variant of the one-dimensional small-world network (SWN). In the network structure considered, the probability of a shortcut falling between a pair of nodes is proportional to r(-alpha) , where r is the lattice distance between the nodes. When alpha=0 , it reduces to the SWN model with random shortcuts. The system shows the dynamic small-world effect, which is different from the well-studied static SW effect. We study the effective network diameter, the path length as a function of the lattice distance, and the dynamics. They are controlled by multiple parameters, and we use data collapse to show that the parameters are correlated. The central finding is that, in the one-dimensional network studied, the dynamic SW effect exists for 0</=alpha</=2 . For each given value of alpha in this region, the point where the dynamic SW effect arises is M L' approximately 1 , where M is the number of useful shortcuts and L' is their average reduced (effective) length.