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Abstract Assessing sensitivity to unmeasured confounding is an important step in observational studies, which typically estimate effects under the assumption that all confounders are measured. In this paper, we develop a sensitivity analysis framework for balancing weights estimators, an increasingly popular approach that solves an optimization problem to obtain weights that directly minimizes covariate imbalance. In particular, we adapt a sensitivity analysis framework using the percentile bootstrap for a broad class of balancing weights estimators. We prove that the percentile bootstrap procedure can, with only minor modifications, yield valid confidence intervals for causal effects under restrictions on the level of unmeasured confounding. We also propose an amplification—a mapping from a one-dimensional sensitivity analysis to a higher dimensional sensitivity analysis—to allow for interpretable sensitivity parameters in the balancing weights framework. We illustrate our method through extensive real data examples. 

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms that perform clustering on several small subgraphs and finally patch the results into a single clustering. The main advantage of these algorithms is that they significantly bring down the computational cost of traditional algorithms, including spectral clustering, semidefinite programs, modularity-based methods, likelihood-based methods, etc., without losing accuracy, and even improving accuracy at times. These algorithms are also, by nature, parallelizable. Since most traditional algorithms are accurate, and the corresponding optimization problems are much simpler in small problems, our divide-and-conquer methods provide an omnibus recipe for scaling traditional algorithms up to large networks. We prove the consistency of these algorithms under various subgraph selection procedures and perform extensive simulations and real-data analysis to understand the advantages of the divide-and-conquer approach in various settings.