# Further Exploration of an Upper Bound for Kemeny’s Constant

## TL;DR

This paper explores a mathematical upper bound for Kemeny’s constant in graphs and shows how it can be used efficiently for large networks.

## Contribution

The paper generalizes previous bounds for specific graph classes and demonstrates practical numerical approximations for large-scale networks.

## Key findings

- The previously found bound is tight for bipartite and windmill graphs.
- Numerical approximations using the bound are efficient for real-world networks.
- The method provides a 30x speedup for graphs with up to 100K nodes.

## Abstract

Even though Kemeny’s constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny’s constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny’s constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time.