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 life time <T>$ of a lamb scales as <T>\sim N$ and the survival probability $S(N\to \infty,t)$ becomes finite on scale-free networks with degree exponent $\gamma>3$. However, $S(N,t)$ for $\gamma<3$ has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. It suggests that the second moment of degree distribution <k^2>$ is the relevant factor for the dynamical properties in diffusive capture process. We numerically find that the normalized number of capture events at a node with degree $k$, $n(k)$, decreases as $n(k)\sim k^{-\sigma}$. When $\gamma<3$, $n(k)$ still increases anomalously for $k\approx k_{max}$. We analytically show that $n(k)$ satisfies the relation $n(k)\sim k^2P(k)$ and the total number of capture events $N_{tot}$ is proportional to <k^2>$, which causes the $\gamma$ dependent behavior of $S(N,t)$ and <T>$.