We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport distance over graph edges. We show that our distance can be computed using a single linear program without precomputing pairwise assignment costs and derive several theoretical properties of the metric. Finally, we provide experiments demonstrating these properties empirically, specifically focusing on its value for new problems in ensemble-based analysis of political districting plans.
@article{arxiv.1910.09618,
title = {Geometry of Graph Partitions via Optimal Transport},
author = {Tara Abrishami and Nestor Guillen and Parker Rule and Zachary Schutzman and Justin Solomon and Thomas Weighill and Si Wu},
journal= {arXiv preprint arXiv:1910.09618},
year = {2019}
}