The k-ary n-cube has many good characteristics， and it has become one of the most commonly used interconnection network topologies in multiprocessor systems. The maintenance ability of system subnetworks plays an important role for the practical applications of the systems when failures occur in the interconnection network. In order to accurately measure the fault tolerance of subnetworks with arbitrary size in a k-ary n-cube， the reliability of k-ary （n-m）-cube subnetworks in a k-ary n-cube in the presence of failures was studied. When k was an odd integer and k was bigger than 2， the upper bound and lower bound on the probability that at least one k-ary （n-m）-cube subnetwork was fault-free in a k-ary n-cube were obtained under the probabilistic fault condition， and an approximate method for evaluating the reliability was proposed. Experimental results show that there is a gradual convergence between the upper bound and lower bound on the k-ary （n-m）-cube subnetwork reliability as the vertex reliability decreases， and the evaluation result obtained by the approximate method is relatively accurate when the vertex reliability is large.

The implementation of the functions of a multiprocessor system relies heavily on the topological properties of the interconnection network of this system. The subnetwork reliability of k-ary n-cube network is an important factor that needs to be taken into account when the computing tasks are processed by the multiprocessor systems constructed with k-ary n-cube as topological structure. In order to accurately and efficiently measure the reliability of the k-ary （n-1）-cube subnetwork in a k-ary n-cube under the probabilistic fault condition， an approximate method to evaluate the reliability of k-ary （n-1）-cube subnetwork based on the Back Propagation （BP） neural network was proposed. Firstly， the generation method for dataset to train BP neural network was given by the aid of the Monte Carlo simulation method and the known upper and lower bounds on the reliability of the k-ary （n-1）-cube subnetwork. Then， the BP neural network model for evaluating the reliability of the k-ary （n-1）-cube subnetwork was constructed on the basis of the generated training dataset. Finally， the approximate evaluation results of the k-ary （n-1）-cube subnetwork reliability obtained by the BP neural network model were analyzed and compared with the results obtained by the approximate calculation formula and the evaluation method based on Monte Carlo simulation. The results obtained by the proposed method were more accurate compared with the approximate calculation formula， and the evaluation time of the proposed method was reduced by about 59% on average compared with the evaluation method based on Monte Carlo simulation. Experimental results show that the proposed method has certain advantages in balancing accuracy and efficiency.