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We consider a quenched contact process in a directed complex networks such as random networks, small-world networks, and scale-free networks. We observed the Griffiths phases which the power law appears in a critical region. In this work we consider the mixture of both undirected and directed links in the complex networks. The direction of the links influenced on the lower bound of criticality. We obtained a phase diagram on a plane of the probability of directional link and the infection rate. We also identify the rare-regions which are induced from the direction of the links.