Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance R_{e e} and the number of distinct visited nodes N_v of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈 R_{e e} 〉 shows the dynamical scaling behavior 〈 R_{e e} (over(ℓ, -), t) 〉 = over(ℓ, -)^α (γ, N) g (t / over(ℓ, -)^z), where over(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, 〈 R_{e e} (over(ℓ, -), t) 〉 in the limit t → ∞ satisfies the relation 〈 R_{e e} 〉 ∼ over(ℓ, -)^α ∼ d^α, where d is the diameter of network with d (over(ℓ, -)) ≃ ln N for γ ≥ 3 or d (over(ℓ, -)) ≃ ln ln N for γ < 3. Based on the scaling relation 〈 R_{e e} 〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.