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Efficient Algorithms for Minimizing the Kirchhoff Index via Adding Edges

Xiaotian Zhou, Ahad N. Zehmakan, Zhongzhi Zhang

TL;DR

This work addresses minimizing the Kirchhoff index by adding $k$ edges to a sparse graph and develops both a deterministic greedy and a gradient-based greedy approach. It introduces near-linear-time methods based on convex-hull approximation, Johnson–Lindenstrauss projection, and fast Laplacian solvers to approximate edge gradients, enabling scalable optimization on graphs with millions of nodes. Theoretical guarantees are provided for the greedy variants via submodularity ratio and curvature bounds, while practical algorithms (Grad, FastGrad, FastGrad+, OneConv) demonstrate strong empirical performance and scalability, outperforming existing state-of-the-art methods in many settings. The results highlight significant potential for efficient network editing to improve global connectivity and robustness in large-scale systems.

Abstract

The Kirchhoff index, which is the sum of the resistance distance between every pair of nodes in a network, is a key metric for gauging network performance, where lower values signify enhanced performance. In this paper, we study the problem of minimizing the Kirchhoff index by adding edges. We first provide a greedy algorithm for solving this problem and give an analysis of its quality based on the bounds of the submodularity ratio and the curvature. Then, we introduce a gradient-based greedy algorithm as a new paradigm to solve this problem. To accelerate the computation cost, we leverage geometric properties, convex hull approximation, and approximation of the projected coordinate of each point. To further improve this algorithm, we use pre-pruning and fast update techniques, making it particularly suitable for large networks. Our proposed algorithms have nearly-linear time complexity. We provide extensive experiments on ten real networks to evaluate the quality of our algorithms. The results demonstrate that our proposed algorithms outperform the state-of-the-art methods in terms of efficiency and effectiveness. Moreover, our algorithms are scalable to large graphs with over 5 million nodes and 12 million edges.

Efficient Algorithms for Minimizing the Kirchhoff Index via Adding Edges

TL;DR

This work addresses minimizing the Kirchhoff index by adding edges to a sparse graph and develops both a deterministic greedy and a gradient-based greedy approach. It introduces near-linear-time methods based on convex-hull approximation, Johnson–Lindenstrauss projection, and fast Laplacian solvers to approximate edge gradients, enabling scalable optimization on graphs with millions of nodes. Theoretical guarantees are provided for the greedy variants via submodularity ratio and curvature bounds, while practical algorithms (Grad, FastGrad, FastGrad+, OneConv) demonstrate strong empirical performance and scalability, outperforming existing state-of-the-art methods in many settings. The results highlight significant potential for efficient network editing to improve global connectivity and robustness in large-scale systems.

Abstract

The Kirchhoff index, which is the sum of the resistance distance between every pair of nodes in a network, is a key metric for gauging network performance, where lower values signify enhanced performance. In this paper, we study the problem of minimizing the Kirchhoff index by adding edges. We first provide a greedy algorithm for solving this problem and give an analysis of its quality based on the bounds of the submodularity ratio and the curvature. Then, we introduce a gradient-based greedy algorithm as a new paradigm to solve this problem. To accelerate the computation cost, we leverage geometric properties, convex hull approximation, and approximation of the projected coordinate of each point. To further improve this algorithm, we use pre-pruning and fast update techniques, making it particularly suitable for large networks. Our proposed algorithms have nearly-linear time complexity. We provide extensive experiments on ten real networks to evaluate the quality of our algorithms. The results demonstrate that our proposed algorithms outperform the state-of-the-art methods in terms of efficiency and effectiveness. Moreover, our algorithms are scalable to large graphs with over 5 million nodes and 12 million edges.

Paper Structure

This paper contains 29 sections, 11 theorems, 26 equations, 3 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Graph and Related Matrices
  5. Effective Resistance and Kirchhoff Index
  6. Concepts Related to Set Function
  7. Problem Formulation
  8. Problem Statement
  9. Deterministic Greedy Algorithm
  10. Reducing Time Complexity.
  11. Evaluating the Greedy Algorithm's Quality
  12. Enhanced Greedy Algorithm via Two Approximation Algorithms
  13. Nearly Linear Time Algorithm
  14. Transforming Decrement Search to Gradient Maximization
  15. Primary Nearly Linear-time Solution
  16. ...and 14 more sections

Key Result

Theorem 3.1

Let $\gamma \in [0,1]$ and $\alpha \in [0,1]$ be submodularity ratio and curvature of the function $f(T) = K(\mathcal{G})- K(T)$ for $T \subseteq Q$. Then, the edge set $T$ returned by Algorithm alg:det satisfies where

Figures (3)

  • Figure 1: Kirchhoff index divided by $n$ returned by all algorithms for $k=1,2,\ldots,50$ on six small networks, BitcoinAlpha (a), EmailUniv (b), Anybeat (c), EmailEU (d), Gnutella08 (e), and Government (f).
  • Figure 2: Kirchhoff index divided by $n$ returned by algorithms FastGrad, FastGrad+, and OneConv, for $k=1,2,\ldots,50$ on four large networks, Douban (a), Delicious (b), YoutubeSnap (c), and DBLP (d).
  • Figure 3: Running time of different algorithms for $k=10,20,\ldots,50$ on four networks, Anybeat (a), EmailEU (b), YoutubeSnap (c), and DBLP (d).

Theorems & Definitions (16)

  • Definition 2.1: Submodularity ratio AbDa11
  • Definition 2.2: Curvature BiBuKrTs17
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 6 more