Random walks and diameter of finite scale-free networks

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