Coupling cyber and physical systems gives rise to numerous engineering challenges and opportunities. An important challenge is the contagion of failure from one system to another, which can lead to large-scale cascading failures. On the other hand, self-healing ability emerges as a valuable opportunity where the overlay cyber network can cure failures in the underlying physical network. To capture both self-healing and contagion, we introduce a factor graph representation of inter-dependent cyber-physical systems, in which factor nodes represent various network functionalities and the edges capture the interactions across different layers. We extend the message-passing algorithm used in low-density parity-check codes to the proposed representation. Through applying the Bayesian rule, we obtain an approximation formula for a given network to investigate the reaction of the network to initial disruption. Our analysis provides simple yet critical guidelines for considering network parameters to achieve resiliency against cascading failures.