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 , where 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, in the limit t→∞ satisfies the relation , where d is the diameter of network with for γ≥3 or 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.