Fast Query of Biharmonic Distance in Networks

TL;DR

This paper introduces efficient algorithms for approximating the biharmonic distance in large graphs, combining locality and provable guarantees, with extensive empirical validation on benchmark networks.

Contribution

The work presents novel algorithms for fast, approximate computation of biharmonic distance with performance guarantees and locality properties.

Findings

Algorithms achieve small additive error in near-logarithmic time

Extensive empirical validation confirms accuracy and efficiency

Applicable to large benchmark networks

Abstract

The \textit{biharmonic distance} (BD) is a fundamental metric that measures the distance of two nodes in a graph. It has found applications in network coherence, machine learning, and computational graphics, among others. In spite of BD's importance, efficient algorithms for the exact computation or approximation of this metric on large graphs remain notably absent. In this work, we provide several algorithms to estimate BD, building on a novel formulation of this metric. These algorithms enjoy locality property (that is, they only read a small portion of the input graph) and at the same time possess provable performance guarantees. In particular, our main algorithms approximate the BD between any node pair with an arbitrarily small additive error $\eps$ in time $O(\frac{1}{\eps^2}\text{poly}(\log\frac{n}{\eps} ))$. Furthermore, we perform an extensive empirical study on several benchmark networks, validating the performance and accuracy of our algorithms.