Efficient Algorithms for Minimizing the Kirchhoff Index via Adding Edges

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...

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Bibliographic Details
Published in:IEEE transactions on knowledge and data engineering Vol. 37; no. 6; pp. 3342 - 3355
Main Authors: Zhou, Xiaotian, Zehmakan, Ahad N., Zhang, Zhongzhi
Format: Journal Article
Language:English
Published: IEEE 01.06.2025
Subjects:
Approximation algorithms
Data mining
Error analysis
graph algorithm
Greedy algorithms
Indexes
Kirchhoff index
Laplace equations
Optimization
Resistance
Resistance distance
Time complexity
Vectors
ISSN:1041-4347, 1558-2191
Online Access:Get full text
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Summary: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.
ISSN:1041-4347
1558-2191
DOI:10.1109/TKDE.2025.3552644