# Biharmonic Distance of Graphs and its Higher-Order Variants: Theoretical Properties with Applications to Centrality and Clustering

arXiv: https://arxiv.org/abs/2406.07574 · HTML: https://mlsanity.com/paper/2406.07574/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 49 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Rethinking the Expressive Power of GNNs via Graph Biconnectivity](https://mlsanity.com/paper/2301.09505/) (2023)
- [Understanding Oversquashing in GNNs through the Lens of Effective Resistance](https://mlsanity.com/paper/2302.06835/) (2023)
- [Affinity-Aware Graph Networks](https://mlsanity.com/paper/2206.11941/) (2022)
- [Graph Clustering using Effective Resistance](https://mlsanity.com/paper/1711.06530/) (2017)
- [Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms](https://mlsanity.com/paper/1708.05959/) (2017)
- [Biharmonic Distance and the Performance of Second-Order Consensus Networks with Stochastic Disturbances](https://mlsanity.com/paper/1709.02790/) (2017)
- [Topology Design for Optimal Network Coherence](https://mlsanity.com/paper/1411.4884/) (2014)

## Abstract (arXiv)

Effective resistance is a distance between vertices of a graph that is both theoretically interesting and useful in applications. We study a variant of effective resistance called the biharmonic distance. While the effective resistance measures how well-connected two vertices are, we prove several theoretical results supporting the idea that the biharmonic distance measures how important an edge is to the global topology of the graph. Our theoretical results connect the biharmonic distance to well-known measures of connectivity of a graph like its total resistance and sparsity. Based on these results, we introduce two clustering algorithms using the biharmonic distance. Finally, we introduce a further generalization of the biharmonic distance that we call the k-harmonic distance. We empirically study the utility of biharmonic and k-harmonic distance for edge centrality and graph clustering.