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