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.

翻译：基尔霍夫指数是网络中所有节点对之间电阻距离之和，是衡量网络性能的关键指标，其值越低表示性能越优。本文研究了通过添加边来最小化基尔霍夫指数的问题。我们首先提出了一种解决该问题的贪心算法，并基于次模性比率和曲率的界限分析了其质量。接着，我们引入了一种基于梯度的贪心算法作为解决该问题的新范式。为加速计算成本，我们利用了几何特性、凸包近似以及各点投影坐标的近似方法。为进一步优化该算法，我们采用了预剪枝和快速更新技术，使其特别适用于大规模网络。我们提出的算法具有近线性的时间复杂度。我们在十个真实网络上进行了大量实验以评估算法质量。结果表明，我们提出的算法在效率和效果上均优于现有最先进方法。此外，我们的算法可扩展至具有超过500万个节点和1200万条边的大规模图。