🍩 Database of Original & Non-Theoretical Uses of Topology

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  1. The (Homological) Persistence of Gerrymandering (2021)

    Moon Duchin, Tom Needham, Thomas Weighill
    Abstract \textlessp style='text-indent:20px;'\textgreaterWe apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. We begin by combining geographic and electoral data from a districting plan to produce a persistence diagram. Then, to see beyond a particular plan and understand the possibilities afforded by the choices made in redistricting, we build methods to visualize and analyze large ensembles of alternative plans. Our detailed case studies use zero-dimensional homology (persistent components) of filtered graphs constructed from voting data to analyze redistricting in Pennsylvania and North Carolina. We find that, across large ensembles of partitions, the features cluster in the persistence diagrams in a way that corresponds strongly to geographic location, so that we can construct an average diagram for an ensemble, with each point identified with a geographical region. Using this localization lets us produce zonings of each state at Congressional, state Senate, and state House scales, show the regional non-uniformity of election shifts, and identify attributes of partitions that tend to correspond to partisan advantage.\textless/p\textgreater\textlessp style='text-indent:20px;'\textgreaterThe methods here are set up to be broadly applicable to the use of TDA on large ensembles of data. Many studies will benefit from interpretable summaries of large sets of samples or simulations, and the work here on localization and zoning will readily generalize to other partition problems, which are abundant in scientific applications. For the mathematically and politically rich problem of redistricting in particular, TDA provides a powerful and elegant summarization tool whose findings will be useful for practitioners.\textless/p\textgreater