Concentration of the Kirchhoff index for Erdos-Renyi graphs

TL;DR

This paper analyzes the Kirchhoff index in Erdős-Rényi graphs, providing formulas for its expected value and showing it concentrates around this expectation, with implications for graph connectivity and measurement strategies in synchronization tasks.

Contribution

It derives the expected Kirchhoff index for Erdős-Rényi graphs and proves its concentration, linking spectral properties to connectivity and measurement efficiency.

Findings

Expected Kirchhoff index formulas for Erdős-Rényi graphs

Concentration results around the expected value

Implications for measurement strategies in synchronization

Abstract

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 Erdos-Renyi 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 Erdos-Renyi 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.