Diffusive capture process on complex networks

We study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of N. We find that the lifetime T of a lamb scales as T ∼N and the survival probability S (N→,t) becomes finite on scale-free networks with degree exponent γ>3. However, S (N,t) for γ<3 has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. This suggests that the second moment of degree distribution k2 is the relevant factor for the dynamical properties in the diffusive capture process. We numerically find that the normalized number of capture events at a node with degree k, n (k), decreases as n (k) ∼ k-σ. When γ<3, n (k) still increases anomalously for k kmax, where kmax is the maximum value of k of given networks with size N. We analytically show that n (k) satisfies the relation n (k) ∼ k2 P (k) for any degree distribution P (k) and the total number of capture events Ntot is proportional to k2 , which causes the γ -dependent behavior of S (N,t) and T.