ProbeWalk: Fast Estimation of Biharmonic Distance on Graphs via Probe-Driven Random Walks

TL;DR

ProbeWalk introduces a faster, probe-driven random walk algorithm for estimating biharmonic distances on large graphs, significantly reducing computational complexity and enabling scalable analysis in real-world networks.

Contribution

It presents a novel algorithm that improves the complexity of biharmonic distance estimation from O(L^5) to O(L^3) with a relative-error guarantee, outperforming existing methods.

Findings

Achieves 10x-1000x speedups over baselines

Scales to graphs with tens of millions of nodes

Provides accurate relative-error estimates

Abstract

The biharmonic distance is a fundamental metric on graphs that measures the dissimilarity between two nodes, capturing both local and global structures. It has found applications across various fields, including network centrality, graph clustering, and machine learning. These applications typically require efficient evaluation of pairwise biharmonic distances. However, existing algorithms remain computationally expensive. The state-of-the-art method attains an absolute-error guarantee epsilon_abs with time complexity O(L^5 / epsilon_abs^2), where L denotes the truncation length. In this work, we improve the complexity to O(L^3 / epsilon^2) under a relative-error guarantee epsilon via probe-driven random walks. We provide a relative-error guarantee rather than an absolute-error guarantee because biharmonic distances vary by orders of magnitude across node pairs. Since L is often very large in real-world networks (for example, L >= 10^3), reducing the L-dependence from the fifth to the third power yields substantial gains. Extensive experiments on real-world networks show that our method delivers 10x-1000x per-query speedups at matched relative error over strong baselines and scales to graphs with tens of millions of nodes.