Abstract
Consider a complex network whose components either work or fail, there probabilities are known. Our paper study two methods for the exact computation of the reliability network, the first one is a decomposition method which may be used even after the community can go through no further modular decomposition, is based on a partition the vertices of network into sets that can be analyzed them sequentially. This method involves choosing one component as a key and then calculate the reliability network twice the first when the key work and the other if the key failed. These two probabilities are then combined to obtain the reliability network. The second method is inclusion-exclusion method which is one of the earliest techniques to compute complex network reliability expressions using the probability laws. We get the same polynomial by using two methods. In both methods, we need to calculate all of the minimum paths so we calculated them by use the adjacency matrix.
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