  Typically, contagion strength is modeled by a transmission rate $\lambda$,
whereby all nodes in a network are treated uniformly in a mean-field
approximation. However, local agents react differently to the same contagion
based on their local characteristics. Following our recent work [EPL
\textbf{99}, 58002 (2012)], we investigate contagion spreading models with
local dynamics on complex networks. We therefore quantify contagions by their
quality, $0 \leq \alpha \leq 1$, and follow their spreading as their
transmission condition (fitness) is evaluated by local agents. We choose
various deterministic local rules. Initial spreading with exponential
quality-dependent time scales is followed by a stationary state with a
prevalence depending on the quality of the contagion. We also observe various
interesting phenomena, for example, high prevalence without the participation
of the hubs. This is in sharp contrast with the general belief that hubs play a
central role in a typical spreading process. We further study the role of
network topology in various models and find that as long as small-world effect
exists, the underlying topology does not contribute to the final stationary
state but only affects the initial spreading velocity.
