In a network system, a propagated failure (PF) is a failure originating from a network component that can cause extensive damages to other network components or even the failure of the entire system. Existing works on PFs have mostly assumed the deterministic effect from a component PF, i.e., a fixed subset of system components is affected whenever the PF occurs. However, in many real-world systems, the components may have different levels of protection, and the effect of damage from a component PF can be dependent upon the status of other components within the same system or the occurrence order of component failures. This paper proposes a new analytical method based on multi-valued decision diagrams (MDDs) for the reliability analysis of network systems with dependent propagation effects. Particularly, new MDD modeling procedures are proposed for considering different types of dependent PF effects introduced by different protection levels. After the system MDD is generated using a new MDD combination algorithm to efficiently handle the dependent PF effects, methods for computing the network reliability and component importance measures are presented. The detailed analysis of an example network system subjected to dependent PFs is presented to illustrate the basics and application of the proposed method. It is shown that the proposed MDD-based method generates smaller model size and thus presents lower computational complexity in the model generation and evaluation than the existing Markov method and separable method.