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The diagonal entries of pseudoinverse of the Laplacian matrix of a graphappear in many important practical applications, since they contain muchinformation of the graph and many relevant quantities can be expressed in termsof them, such as Kirchhoff index and current flow centrality. However, ana\"{\i}ve approach for computing the diagonal of a matrix inverse has cubiccomputational complexity in terms of the matrix dimension, which is notacceptable for large graphs with millions of nodes. Thus, rigorous solutions tothe diagonal of the Laplacian matrices for general graphs, even for particluargraphs are much less. In this paper, we propose a theoretically guaranteedestimation algorithm, which approximates all diagonal entries of thepseudoinverse of a graph Laplacian in nearly linear time with respect to thenumber of edges in the graph. We execute extensive experiments on real-lifenetworks, which indicate that our algorithm is both efficient and accurate.Also, we determine exact expressions for the diagonal elements of pseudoinverseof the Laplacian matrices for Koch networks and uniform recursive trees, andcompare them with those obtained by our approximation algorithm. Finally, weuse our algorithm to evaluate the Kirchhoff index of three deterministic modelnetworks, for which the Kirchhoff index can be rigorously determined. Theseresults further show the effectiveness and efficiency of our algorithm.

Diagonal of Pseudoinverse of Graph Laplacian: Fast Estimation and Exact Results