The diagonal entries of pseudoinverse of the Laplacian matrix of a graph appear in many important practical applications, since they contain much information of the graph and many relevant quantities can be expressed in terms of them, such as Kirchhoff index and current flow centrality. However, a na\"{\i}ve approach for computing the diagonal of a matrix inverse has cubic computational complexity in terms of the matrix dimension, which is not acceptable for large graphs with millions of nodes. Thus, rigorous solutions to the diagonal of the Laplacian matrices for general graphs, even for particluar graphs are much less. In this paper, we propose a theoretically guaranteed estimation algorithm, which approximates all diagonal entries of the pseudoinverse of a graph Laplacian in nearly linear time with respect to the number of edges in the graph. We execute extensive experiments on real-life networks, which indicate that our algorithm is both efficient and accurate. Also, we determine exact expressions for the diagonal elements of pseudoinverse of the Laplacian matrices for Koch networks and uniform recursive trees, and compare them with those obtained by our approximation algorithm. Finally, we use our algorithm to evaluate the Kirchhoff index of three deterministic model networks, for which the Kirchhoff index can be rigorously determined. These results further show the effectiveness and efficiency of our algorithm.

翻译：图拉普拉斯矩阵伪逆的对角线元素在许多重要实际应用中具有关键作用，因为它们包含了图的丰富信息，并且许多相关量（如基尔霍夫指数和电流中心性）可据此表达。然而，直接计算矩阵逆对角线的朴素方法具有与矩阵维度相关的三次计算复杂度，对于包含数百万节点的大规模图而言不可接受。因此，针对一般图甚至特定图拉普拉斯矩阵的严格解仍较为稀缺。本文提出一种具有理论保证的估计算法，能够以接近图边数线性时间近似计算拉普拉斯矩阵伪逆的所有对角线元素。我们在真实网络上的大量实验表明，该算法兼具高效性与准确性。此外，我们确定了科赫网络和均匀递归树拉普拉斯矩阵伪逆对角线元素的精确表达式，并将其与近似算法结果进行对比。最后，我们运用该算法评估三个确定性模型网络的基尔霍夫指数（该指数可严格确定），结果进一步验证了算法的有效性与高效性。