  The Contact Process has been studied on complex networks exhibiting different
kinds of quenched disorder. Numerical evidence is found for Griffiths phases
and other rare region effects, in Erd\H os R\'enyi networks, leading rather
generically to anomalously slow (algebraic, logarithmic,...) relaxation. More
surprisingly, it turns out that Griffiths phases can also emerge in the absence
of quenched disorder, as a consequence of sole topological heterogeneity in
networks with finite topological dimension. In case of scale-free networks,
exhibiting infinite topological dimension, slow dynamics can be observed on
tree-like structures and a superimposed weight pattern. In the infinite size
limit the correlated subspaces of vertices seem to cause a smeared phase
transition. These results have a broad spectrum of implications for propagation
phenomena and other dynamical process on networks and are relevant for the
analysis of both models and empirical data.
