Detecting a small number of outliers from a set of data observations is always challenging. This problem is more difficult in the setting of multiple network samples, where computing the anomalous degree of a network sample is generally not sufficient. In fact, explaining why a given network is exceptional, expressed in the form of subnetwork, is also equally important. We develop a novel algorithm to address these two key problems. We treat each network sample as a potential outlier and identify subnetworks that help discriminate it from nearby samples. The algorithm is developed in the framework of network regression combined with the constraints on both network topology and L1-norm shrinkage to perform subnetwork discovery. Our method thus goes beyond subspace/subgraph discovery. We also show that the developed method converges to a global optimum. Empirical evaluation on various real-world network datasets demonstrates the advantages of our algorithm over various baseline methods.