[Paper Review] Geometry of Graph Partitions via Optimal Transport

Tara Abrishami, Nestor Guillen+5 more|arXiv (Cornell University)|Oct 21, 2019

Advanced Graph Theory Research31 references4 citations

TL;DR

This paper introduces a novel optimal transport-based distance metric for graph partitions, computing similarity between partitions via a linear assignment of components using edge-based transport costs. The method enables efficient computation via a single linear program and demonstrates strong utility in visualizing and analyzing political redistricting ensembles, revealing geographically and politically representative plans more effectively than prior methods.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of navigating and analyzing large ensembles of graph partitions, especially in political redistricting, where traditional visualization methods are limited.
• To develop a geometric distance metric between partitions that captures structural and spatial relationships beyond simple overlap or statistical summaries.
• To enable practical, interpretable visualization of partition ensembles using embedding techniques like multidimensional scaling (MDS).
• To provide a theoretically grounded, computationally feasible alternative to naive distance measures such as Hamming or total variation distances.
• To demonstrate the metric's utility in identifying representative and outlier redistricting plans in real-world and synthetic datasets.

Proposed method

• The method defines a distance between partitions by solving a linear assignment problem that matches components of two partitions based on transport cost over graph edges.
• Transport cost between components is computed via the 1-Wasserstein distance on the underlying graph, modeling flow between nodes.
• The overall distance is formulated as a single linear program, avoiding precomputation of pairwise assignment costs.
• The approach is extended to unbalanced partitions through weighted component matching, enabling application to unequal-sized components.
• The metric is invariant to component ordering and sensitive to geometric structure, unlike overlap-based measures such as KL-divergence or total variation.
• The framework supports embedding of partition ensembles into low-dimensional Euclidean space using MDS for visualization and analysis.

Experimental results

Research questions

• RQ1 Can a geometric distance metric be constructed between graph partitions that reflects spatial and structural relationships rather than just overlap?
• RQ2 How can optimal transport be hierarchically applied to define a meaningful distance on the space of partitions?
• RQ3 To what extent does the proposed metric improve the visualization and interpretability of large ensembles of redistricting plans?
• RQ4 How does the transport-based distance compare to traditional metrics like Hamming or total variation in capturing representativeness and clustering in partition spaces?
• RQ5 Can the metric identify geographically and politically representative plans more effectively than existing methods?

Key findings

• The proposed metric enables efficient computation of partition distances via a single linear program, significantly reducing computational overhead compared to iterative assignment methods.
• In both synthetic and real-world redistricting data, the transport-based distance successfully identified the judges' plan as central and representative within the ensemble, while the 2012 and 2016 plans were located on the periphery.
• The MDS embeddings generated using the transport metric revealed clear clustering and spatial structure in partition ensembles, demonstrating its value for visual navigation and analysis.
• The metric demonstrated robustness to variations in ensemble generation methods, consistently placing the judges’ plan near the center of the embedding regardless of whether the ensemble was generated via ReCom or another method.
• Theoretical properties such as invariance to component ordering and sensitivity to geometry were validated empirically, confirming the metric’s intuitive and geometric soundness.
• The framework generalizes to unbalanced partitions through weighted component matching, broadening its applicability to real-world scenarios with unequal district sizes.