  We investigate the dynamic scaling properties of stochastic particle systems
on a non-deterministic scale-free network. It has been known that the dynamic
scaling behavior depends on the degree distribution exponent of the underlying
scale-free network. Our study shows that it also depends on the global
structure of the underlying network. In random walks on the tree structure
scale-free network, we find that the relaxation time follows a power-law
scaling $\tau\sim N$ with the network size $N$. And the random walker return
probability decays algebraically with the decay exponent which varies from node
to node. On the other hand, in random walks on the looped scale-free network,
they do not show the power-law scaling. We also study a pair-annihilation
process on the scale-free network with the tree and the looped structure,
respectively. We find that the particle density decays algebraically in time
both cases, but with the different exponent.
