Project leader: Professor Gregory Herschlag and Professor Biji Wong
Project manager: Spencer Whitehead
Team members: Jenny Chan, Peakay Clifford, Eileen Santana, Felix Sesin, Leo Yang
Past attempts at quantifying redistricting have often relied on usual statistical methods; for instance, one may generate a large sample, or ensemble, of districting plans on the map of precincts in a US state (such as CT) and compute the number of congressional districts won by republicans and democrats according to real election data. Such methods identify outlier districting plans whose partisan outcomes are statistically unlikely, providing evidence of possible gerrymandering. This is how gerrymandering may be detected at the global level, that is, across a districting plan. In this project, we seek to develop tools which enable this sort of statistical analysis and detection of gerrymandering on a more local level – at the level of individual districts in a districting plan. The essential idea is the following: to determine whether a district votes atypically, we must first learn how districts similar to the enacted district vote,
and this requires learning exactly which districts are similar to the enacted district in the first place.
Over the past twenty years, classical diffusion has been developed as a statistical method for embedding data objects in Euclidean space, with similar data objects clustering close to each other. This is precisely what enables learning similarity between districts. However, when individual data objects contain internal structure of interest, classical diffusion fails to capture the full picture. More recently, Gao (2019) proposed horizontal diffusion maps (HDMs), a new framework which does consider this internal structure, and he showed that HDMs clustered lemur teeth more effectively compared to classical diffusion. HDM considers diffusion on the total space of data points within the data objects, which are the discrete or discretized components of the internal structure of each data object. In order to consider the internal precinct structure of congressional districts, we apply the HDM framework to the redistricting problem.Then, to correct for potentially nonuniform sampling of districts and focus diffusion more heavily on the geometric centers of districts, weighted by population, we propose the explicit construction of a stationary distribution on the total space and enforce it on the HDM operator by the Metropolis-Hastings algorithm. Lastly, we apply this biased HDM operator to ensembles on toy examples and Connecticut data to investigate how effectively districts can be clustered.
Gao, T. (2019). The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps (arXiv:1602.02330).
Eigenmodes of horizontal diffusion on the CT districting space
Eigenmodes of horizontal diffusion on the CT districting space
Ensemble of NC districting plans for obtaining districting space
Ensemble of NC districting plans for obtaining districting space
Vertical diffusion is walking between precincts and horizontal diffusion is jumping between district