Summary: The subnetwork reliability of the interconnection network of a multiprocessor system is an important factor for processing the computing tasks. In order to measure reliability of the arbitrary size subnetwork of the \((n, k)\)-arrangement graph \(A_{n , k}\), the probability \(R_{n , k}^m(p)\) of existing fault-free \(A_{n - m , k - m}\) subnetworks in \(A_{n , k}\) under the probabilistic fault condition is studied for \(m \geq 1\). An upper bound and a lower bound of \(R_{n , k}^m(p)\) are derived, and a heuristic algorithm based on the Monte Carlo simulation to calculate \(R_{n , k}^m(p)\) is proposed. Experimental results show that the upper and lower bounds of \(R_{n , k}^m(p)\) are in accordance with the results of the heuristic algorithm when the upper and lower bounds are close; otherwise, the results of the heuristic algorithm are relatively accurate. Under the consideration that high accuracy requirement and large size of \(A_{n , k}\) may greatly reduce the evaluation efficiency when the heuristic algorithm is used, the back propagation neural network model for evaluating \(R_{n , k}^m(p)\) is constructed, and the experiments on the generated data indicate that the constructed model has certain advantages in balancing accuracy and efficiency.