  Quenched disorder is known to play a relevant role in dynamical processes and
phase transitions. Its effects on the dynamics of complex networks have hardly
been studied. Aimed at filling this gap, we analyze the Contact Process, i.e.
the simplest propagation model, with quenched disorder on complex networks. We
find Griffiths phases and other rare region effects, leading rather generically
to anomalously slow (algebraic, logarithmic, ...) relaxation, on Erd\H
os-R\'enyi networks. Similar effects are predicted to exist for other
topologies with a finite percolation threshold. More surprisingly, we find that
Griffiths phases can also emerge in the absence of quenched disorder, as a
consequence of topological heterogeneity in networks with finite topological
dimension. These results have a broad spectrum of implications for propagation
phenomena and other dynamical processes on networks.
