The power domination problem seeks to determine the minimum number of phasor measurement units (PMUs) needed to monitor an electric power network. We introduce random sensor failure before the power domination process occurs and call this the fragile power domination process. For a given graph, PMU placement, and probability of PMU failure $q$, we study the expected number of observed vertices at the termination of the fragile power domination process. This expected value is a polynomial in $q$, which we relate to fault-tolerant and PMU-defect-robust power domination. We also study the probability of that the entire graph becomes observed and give results for some graph families.

Recent efforts in interpretable deep learning models have shown that concept-based explanation methods achieve competitive accuracy with standard end-to-end models and enable reasoning and intervention about extracted high-level visual concepts from images, e.g., identifying the wing color and beak length for bird-species classification. However, these concept bottleneck models rely on a necessary and sufficient set of predefined concepts-which is intractable for complex tasks such as video classification. For complex tasks, the labels and the relationship between visual elements span many frames, e.g., identifying a bird flying or catching prey-necessitating concepts with various levels of abstraction. To this end, we present CoDEx, an automatic Concept Discovery and Extraction module that rigorously composes a necessary and sufficient set of concept abstractions for concept-based video classification. CoDEx identifies a rich set of complex concept abstractions from natural language explanations of videos-obviating the need to predefine the amorphous set of concepts. To demonstrate our method's viability, we construct two new public datasets that combine existing complex video classification datasets with short, crowd-sourced natural language explanations for their labels. Our method elicits inherent complex concept abstractions in natural language to generalize concept-bottleneck methods to complex tasks.