The pseudo-inverse of a graph Laplacian matrix, denoted as $L^\dagger$, finds extensive application in various graph analysis tasks. Notable examples include the calculation of electrical closeness centrality, determination of Kemeny's constant, and evaluation of resistance distance. However, existing algorithms for computing $L^\dagger$ are often computationally expensive when dealing with large graphs. To overcome this challenge, we propose novel solutions for approximating $L^\dagger$ by establishing a connection with the inverse of a Laplacian submatrix $L_v$. This submatrix is obtained by removing the $v$-th row and column from the original Laplacian matrix $L$. The key advantage of this connection is that $L_v^{-1}$ exhibits various interesting combinatorial interpretations. We present two innovative interpretations of $L_v^{-1}$ based on spanning trees and loop-erased random walks, which allow us to develop efficient sampling algorithms. Building upon these new theoretical insights, we propose two novel algorithms for efficiently approximating both electrical closeness centrality and Kemeny's constant. We extensively evaluate the performance of our algorithms on five real-life datasets. The results demonstrate that our novel approaches significantly outperform the state-of-the-art methods by several orders of magnitude in terms of both running time and estimation errors for these two graph analysis tasks. To further illustrate the effectiveness of electrical closeness centrality and Kemeny's constant, we present two case studies that showcase the practical applications of these metrics.

翻译：图拉普拉斯矩阵的伪逆，记为$L^\dagger$，在各种图分析任务中具有广泛应用。典型例子包括计算电气紧密度中心性、确定凯梅尼常数以及评估电阻距离。然而，现有计算$L^\dagger$的算法在处理大规模图时通常计算成本高昂。为克服这一挑战，我们提出通过建立$L^\dagger$与拉普拉斯子矩阵$L_v$的逆之间的联系来近似$L^\dagger$的新颖解决方案。该子矩阵是通过从原始拉普拉斯矩阵$L$中删除第$v$行和第$v$列得到的。这一联系的关键优势在于$L_v^{-1}$具有多种有趣的组合解释。我们基于生成树和环消随机游走提出了$L_v^{-1}$的两种创新解释，从而得以开发高效的采样算法。基于这些全新的理论洞见，我们提出了两种用于高效近似电气紧密度中心性和凯梅尼常数的新型算法。我们在五个真实数据集上全面评估了算法的性能。结果表明，对于这两项图分析任务，我们提出的新方法在运行时间和估计误差方面均显著优于现有最优方法数个数量级。为进一步说明电气紧密度中心性和凯梅尼常数的有效性，我们通过两个案例研究展示了这些指标的实际应用。