  We provide numerical evidence for slow dynamics of the
susceptible-infected-susceptible model evolving on finite-size random networks
with power-law degree distributions. Extensive simulations were done by
averaging the activity density over many realizations of networks. We
investigated the effects of outliers in both highly fluctuating (natural
cutoff) and non-fluctuating (hard cutoff) most connected vertices. Logarithmic
and power-law decays in time were found for natural and hard cutoffs,
respectively. This happens in extended regions of the control parameter space
$\lambda_1<\lambda<\lambda_2$, suggesting Griffiths effects, induced by the
topological inhomogeneities. Optimal fluctuation theory considering
sample-to-sample fluctuations of the pseudo thresholds is presented to explain
the observed slow dynamics. A quasistationary analysis shows that response
functions remain bounded at $\lambda_2$. We argue these to be signals of a
smeared transition. However, in the thermodynamic limit the Griffiths effects
loose their relevancy and have a conventional critical point at $\lambda_c=0$.
Since many real networks are composed by heterogeneous and weakly connected
modules, the slow dynamics found in our analysis of independent and finite
networks can play an important role for the deeper understanding of such
systems.
