Sum-Product Networks (SPN) are deep probabilistic models with demonstrated excellent performance in several machine learning tasks. As with many other probabilistic models, performing Maximum-A-Posteriori inference in SPNs is NP-hard. Selective SPNs are a subclass of SPNs that allow for efficient Maximum-A-Posteriori inference and closed-form parameter learning. Due to the high number of parameters, SPNs learned from data can produce unreliable and overconfident inferences, especially for instances with low statistical support. This issue can be partially mitigated by performing a robustness analysis of inferences with respect to small changes in the parameters. In this work, we address the problem of assessing the robustness of Maximum-A-Posteriori inferences produced with Selective SPNs to global perturbations of the parameters. We consider such an inference robust if it remains the single maximizer under small perturbations of the model parameters. We present efficient algorithms and an empirical analysis with realistic problems involving missing data completion and multilabel classification. The experiments show that our criteria are informative with respect to the inference accuracy, suggesting that it indeed discriminate robust and non-robust instances.