Random walks and diameter of finite scale-free networks

Abstract

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. $⟨R_{ee}⟩$ shows the dynamical scaling behavior $⟨R_{ee}(\bar{\ell },t)⟩={\bar{\ell }}^{\alpha }(\gamma ,N)g(t/{\bar{\ell }}^z)$, 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, $⟨R_{ee}(\bar{\ell },t)⟩$ in the limit $t\to \infty$ satisfies the relation $⟨R_{ee}⟩\sim {\bar{\ell }}^{\alpha }\sim d^{\alpha }$, 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 $⟨R_{ee}⟩$, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.

arXiv comment4 pages, 4 figures

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