Navigation in a small world with local information

TL;DR

This paper investigates how message navigation occurs in one-dimensional small-world networks with local information, revealing the conditions under which dynamic small-world effects emerge based on network parameters.

Contribution

It introduces a model of navigation in a small-world network with a power-law shortcut distribution and characterizes the dynamic small-world effect for 0 ≤ α ≤ 2.

Findings

Dynamic SW effect exists for 0 ≤ α ≤ 2

Effective network diameter depends on multiple parameters

The onset of the SW effect occurs when ML' ~ 1

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 theoretically study this issue, 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 (SW) 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 $0\leq \alpha \leq 2$. For each given value of $\alpha $ in this region, the point that the dynamic SW effect arises is $ML^{\prime}\sim 1$, where $M$ is the number of useful shortcuts and $L^{\prime}$ is the average reduced (effective) length of them.