Developing a Comprehensive and Coherent Shape Compactness Metric for Gerrymandering

Measuring the form and structure of geographic landscapes is fundamental to understanding geospatial phenomena and their dynamics. Shape compactness metrics have been extensively employed in gerrymandering assessment, urban planning, landscape ecology, and other applications. Existing compactness measurements for gerrymandering, however, target particular aspects of gerrymandered shapes such as elongation, indentation, or dispersion without adequately integrating them with spatial context. This article proposes a comprehensive shape compactness metric that coherently integrates these aspects. It first divides a district into nonoverlapping maximum inscribed circles. Then Euclidean distances from the centers of these circles to the district centroid are standardized and regulated using contextual and topological factors like the distribution of attributes, the fixed upper level boundary, and the relationships with other districts. Finally, these regulated distances are aggregated to produce a quantitative compactness measure that, unlike most existing ones, features a threshold for gerrymandering identification and the coherent integration of roundness, convexity, closeness, and spatial context. Applying the new metric to the U.S. Congressional districts exemplifies its differences from existing metrics and also illustrates the necessity and value of coherently combining multiple aspects of gerrymandering in a single shape compactness metric.