We consider navigation or search schemes on networks which have a degree distribution of the form $P(k) \propto$ exp($−k^{\gamma}$). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter 𝜆. The searches are realistic in the sense that not all search chains can be completed. An estimate of $\mu = \rho/s_{d}$, where 𝜌 is the success rate and $s_{d}$ the dynamic path length, shows that for a network of 𝑁 nodes, $\mu \propto N^{-\delta}$ in general. Dynamic small world effect, i.e., $\delta \simeq 0$ is shown to exist in a restricted region of the $\lambda - \gamma$ plane.