Concentration of the Kirchhoff index for Erdős-Rényi graphs

Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdős-Rényi random graphs. Since the graphs are random, their Kirchhoff indices are random variables. We give formulas for the expected value of the Kirchhoff index and show it concentrates around its expectation. We achieve this by studying the trace of the pseudoinverse of the Laplacian of Erdős-Rényi graphs. For synchronization (a class of estimation problems on graphs) our results imply that acquiring pairwise measurements uniformly at random is a good strategy, even if only a vanishing proportion of the measurements can be acquired.