  Understanding how contagions (information, infections, etc) are spread on
complex networks is important both from practical as well as theoretical point
of view. Considerable work has been done in this regard in the past decade or
so. However, most models are limited in their scope and as a result only
capture general features of spreading phenomena. Here, we propose and study a
model of spreading which takes into account the strength or quality of
contagions as well as the local (probabilistic) dynamics occurring at various
nodes. Transmission occurs only after the quality-based fitness of the
contagion has been evaluated by the local agent. The model exhibits
quality-dependent exponential time scales at early times leading to a slowly
evolving quasi-stationary state. Low prevalence is seen for a wide range of
contagion quality for arbitrary large networks. We also investigate the
activity of nodes and find a power-law distribution with a robust exponent
independent of network topology. Our results are consistent with recent
empirical observations.
