Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance $R_{ee}$ and the number of distinct visited nodes $N_v$ of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. $\left< R_{ee} \right>$ shows the dynamical scaling behavior $\left<R_{ee}({\bar \ell},t)\right>= \bar{\ell}^\alpha (\gamma, N) g(t/\bar{\ell}^z)$, where $\bar{\ell}$ is the average minimum distance between all possible pairs of nodes in the network, $N$ is the number of nodes, $\gamma$ is the degree exponent of the SFN and $t$ is the step number of RWs. Especially, $\left<R_{ee}({\bar \ell},t)\right>$ in the limit $t \to \infty$ satisfies the relation $\left< R_{ee} \right> \sim \bar{\ell}^\alpha \sim d^\alpha$, where $d$ is the diameter of network with $d ({\bar \ell}) \simeq \ln N$ for $\gamma \ge 3$ or $d ({\bar \ell}) \simeq \ln \ln N$ for $\gamma < 3$. Based on the scaling relation $\left< R_{ee} \right>$, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.