Abstract
We study the dynamical properties of a diffusing lamb captured by
a diffusing lion on the scale-free networks with various sizes of
$N$. We find that the life time $łeft<T\right>$ of a lamb scales
as $łeft<T\right>N$ and the survival probability
$S(Nınfty,t)$ 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 $łeft<k^2\right>$
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
$n(k)k^-\sigma$. When $\gamma<3$, $n(k)$ still increases
anomalously for $kk_max$, where $k_max$ is the maximum
value of $k$ of given networks with size $N$. We analytically show
that $n(k)$ satisfies the relation $n(k)k^2P(k)$ for any
degree distribution $P(k)$ and the total number of capture events
$N_tot$ is proportional to $łeft<k^2\right>$, which causes the
$\gamma$ dependent behavior of $S(N,t)$ and $łeft<T\right>$
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