Urban spaces, though often perceived as discrete communities, are shared by various functional and social groups. Our study introduces a graph-based physics-aware deep learning framework, illuminating the intricate overlapping nature inherent in urban communities. Through analysis of individual mobile phone positioning data at Twin Cities metro area (TCMA) in Minnesota, USA, our findings reveal that 95.7 % of urban functional complexity stems from the overlapping structure of communities during weekdays. Significantly, our research not only quantifies these overlaps but also reveals their compelling correlations with income and racial indicators, unraveling the complex segregation patterns in U.S. cities. As the first to elucidate the overlapping nature of urban communities, this work offers a unique geospatial perspective on looking at urban structures, highlighting the nuanced interplay of socioeconomic dynamics within cities.

This paper proposes an easy-to-compute upper bound for the overlap index between two probability distributions without requiring any knowledge of the distribution models. The computation of our bound is time-efficient and memory-efficient and only requires finite samples. The proposed bound shows its value in one-class classification and domain shift analysis. Specifically, in one-class classification, we build a novel one-class classifier by converting the bound into a confidence score function. Unlike most one-class classifiers, the training process is not needed for our classifier. Additionally, the experimental results show that our classifier can be accurate with only a small number of in-class samples and outperform many state-of-the-art methods on various datasets in different one-class classification scenarios. In domain shift analysis, we propose a theorem based on our bound. The theorem is useful in detecting the existence of domain shift and inferring data information. The detection and inference processes are both computation-efficient and memory-efficient. Our work shows significant promise toward broadening the applications of overlap-based metrics.