  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.
