  In this work we investigate the dynamics of random walk processes on
scale-free networks in a short to moderate time scale. We perform extensive
simulations for the calculation of the mean squared displacement, the network
coverage and the survival probability on a network with a concentration $c$ of
static traps. We show that the random walkers remain close to their origin, but
cover a large part of the network at the same time. This behavior is markedly
different than usual random walk processes in the literature. For the trapping
problem we numerically compute $\Phi(n,c)$, the survival probability of mobile
species at time $n$, as a function of the concentration of trap nodes, $c$.
Comparison of our results to the Rosenstock approximation indicate that this is
an adequate description for networks with $2<\gamma<3$ and yield an exponential
decay. For $\gamma>3$ the behavior is more complicated and one needs to employ
a truncated cumulant expansion.
