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

TL;DR

This work introduces the biharmonic distance as a graph distance that better reflects global topology than traditional measures like effective resistance, and it extends to a family of $k$-harmonic distances. It establishes connections between $B_e$ and all-pairs electrical flows via $n\,w_e\,B_e^2 = \sum_{s,t} f_{st}(e)^2/w_e$, derives a down-Laplacian formula $w_e B_e^2=(L^{down})^{+}_{ee}$, and links centrality and sparsity to these distances, including a biharmonic Foster-type theorem. The paper also introduces two clustering algorithms (biharmonic $k$-means and biharmonic Girvan-Newman) and a low-rank approximation for $k$-harmonic distances, with extensive experiments showing superior clustering performance of biharmonic-based methods and strong centrality signals, especially for larger $k$ values. Overall, the biharmonic framework enhances edge-centric analysis and graph clustering, offering scalable computation via Laplacian-based solvers and random projections.

Abstract

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.

Figures (9)

• Figure 1: The effective resistance vs. squared biharmonic distance for edges in a tree. While the effective resistance is 1 on all edges, the biharmonic distance is higher on edges closer to the root, demonstrating that biharmonic distance is aware of the global topology of a graph. See \ref{['apx:examples']} for more examples.
• Figure 2: From right to left: the vector $1_s-1_t$, the electrical flow $f_{st} = \partial^+(1_s - 1_t)$, the potentials $p_{st}=L^{+}(1_s-1_t)$, and the maps that connect them.
• Figure 3: \ref{['thm:sparse_cut_implies_large_biharmonic', 'thm:large_biharmonic_edge_implies_sparse_cut']} suggest that edges crossing sparse cuts will have high biharmonic distance, which we can see in this example of a block stochastic graph.
• Figure 4: Left: The Spearman Rank Correlation Coefficient between different edge centrality measures on the 25-nearest neighbor graph of the Cancer dataset. Here BH is biharmonic distance, CF is current-flow centrality, B is betweenness centrality, ER is effective resistance, and 5H is 5-harmonic distance. Right: The Spearman Rank Correlation Correlation between an edge centrality measure and the same edge centrality measure after a number of random edges were added. Experiments were repeated 5 times for each centrality measure. Results for more graphs can be found in \ref{['sec:centrality_experiments']}.
• Figure 5: Plots of Purity vs. $k$ for $k$-harmonic clustering algorithms. Different plots correspond to nearest neighbor graphs of different datasets. Algorithms not parameterized by $k$ (i.e. Spectral Clustering and Girvan-Newman) are denoted with a dashed line. $k$-harmonic $k$-means, Low Rank $k$-harmonic $k$-means, and Spectral Clustering are averaged over 10 runs of $k$-means with different seeds.

Theorems & Definitions (47)

• Theorem 1.1: GhoshBoydSaber08MinEffRes
• Theorem 1.2: Various Authors
• Lemma 2.0
• Lemma 2.0
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2: Foster's Theorem foster1949average
• Corollary 4.3: Biharmonic Foster's Theorem
• Theorem 5.1
• Theorem 5.2: Folklore