Efficient Algorithms for Minimizing the Kirchhoff Index via Adding Edges

2025-03-27

IEEE Transactions on Knowledge and Data Engineering

AI Summary

This paper addresses the NP-hard problem of minimizing the Kirchhoff index—i.e., the sum of effective resistances between all node pairs—by adding a fixed number of edges to enhance network connectivity and robustness. We propose the first gradient-driven greedy algorithm grounded in submodular ratio and curvature analysis, introducing the novel “gradient heuristic + geometric acceleration” paradigm. Our method integrates convex-hull approximation, projected coordinate estimation, and dynamic resistance distance updates to achieve near-linear time complexity. It supports both pre-pruning and incremental maintenance, significantly outperforming state-of-the-art approaches. Extensive evaluation on ten real-world networks validates its effectiveness, and scalability experiments confirm applicability to massive graphs with over 5 million nodes and 12 million edges. The algorithm achieves both high accuracy and computational efficiency.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• Minimize Kirchhoff index by adding edges to networks
• Develop efficient greedy algorithms for large-scale networks
• Improve computational speed using geometric approximations

Innovation

• Greedy algorithm with submodularity and curvature analysis
• Gradient-based greedy algorithm with geometric acceleration
• Pre-pruning and fast update for large networks