Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance ReeRee and the number of distinct visited nodes NvNv of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈Ree〉〈Ree〉 shows the dynamical scaling behavior 〈Ree(ℓ¯,t)〉=ℓ¯α(γ,N)g(t/ℓ¯z), where ℓ¯ is the average minimum distance between all possible pairs of nodes in the network, NN is the number of nodes, γγ is the degree exponent of the SFN and tt is the step number of RWs. Especially, 〈Ree(ℓ¯,t)〉 in the limit t→∞t→∞ satisfies the relation 〈Ree〉∼ℓ¯α∼dα, where dd is the diameter of network with d(ℓ¯)≃lnN for γ≥3γ≥3 or d(ℓ¯)≃lnlnN for γ<3γ<3. Based on the scaling relation 〈Ree〉〈Ree〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.