Realistic searches on stretched exponential networks

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 $\lambda$. The searches are realistic in the sense that not all search chains can be completed. An estimate of $\mu =\rho \mathrm{/}{s}_d$, where $\rho$ 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.