Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast Query of Biharmonic Distance in Networks

Changan Liu, Ahad N. Zehmakan, Zhongzhi Zhang

TL;DR

This work addresses the scalable computation of biharmonic distance (BD) on large graphs by deriving a new BD formulation via the Laplacian pseudoinverse expansion and developing a family of local algorithms. The Push family (Push, Push+) provides deterministic, locality-driven BD estimation with a universal or node-specific truncation length, while STW and SWF introduce random-walk-based approximations with provable error guarantees and variance-aware stopping. For nodal BD, the SNB and SNB+ methods extend the pairwise approach using summation-estimation to dramatically reduce cost. Empirical results on real networks show substantial speedups over exact and projection-based methods while maintaining accurate BD estimates, enabling practical BD queries in large-scale graphs.

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.

Fast Query of Biharmonic Distance in Networks

TL;DR

This work addresses the scalable computation of biharmonic distance (BD) on large graphs by deriving a new BD formulation via the Laplacian pseudoinverse expansion and developing a family of local algorithms. The Push family (Push, Push+) provides deterministic, locality-driven BD estimation with a universal or node-specific truncation length, while STW and SWF introduce random-walk-based approximations with provable error guarantees and variance-aware stopping. For nodal BD, the SNB and SNB+ methods extend the pairwise approach using summation-estimation to dramatically reduce cost. Empirical results on real networks show substantial speedups over exact and projection-based methods while maintaining accurate BD estimates, enabling practical BD queries in large-scale graphs.

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 in time . Furthermore, we perform an extensive empirical study on several benchmark networks, validating the performance and accuracy of our algorithms.
Paper Structure (23 sections, 12 theorems, 25 equations, 4 figures, 1 table, 4 algorithms)

This paper contains 23 sections, 12 theorems, 25 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Problem Definition
  5. Existing Algorithms
  6. The push algorithm
  7. New Formula for Biharmonic Distance
  8. A Universal Bound of Truncated Length
  9. Algorithm Description
  10. Refining Maximum Length
  11. Approximating by sampling random walks
  12. Approximating by Sampling Random Walks
  13. Full Description of Algorithm
  14. Sampling with Feedback
  15. Nodal Biharmonic Distance Query
  16. ...and 8 more sections

Key Result

Lemma 3.1

Let $\boldsymbol{\mathit{h}}=\sum_{i=0}^{\infty}\boldsymbol{\mathit{b}}_{st}^{\top}\boldsymbol{\mathit{P}}^i\boldsymbol{\mathit{D}}^{-1}$ where $s,t\in V$ are any two distinct nodes, then

Figures (4)

  • Figure 1: Running time of each algorithm on five datasets: (a) Facebook, (b) DBLP, (c) Youtube, (d) Orkut and (e) Livejournal.
  • Figure 2: Performance comparison for approximating nodal BD on two networks: (a) (c) Facebook, (b) (d) DBLP.
  • Figure 3: $\ell$ versus $\ell_{s,t}$ when $\epsilon=0.01$.
  • Figure 4: Absolute error versus $\epsilon$ on five datasets: (a) Facebook, (b) DBLP, (c) Youtube, (d) Orkut and (e) Livejournal.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Lemma 6.1
  • Lemma 6.2
  • ...and 4 more