# The Derivative of Kemeny's Constant as a Centrality Measure in Undirected Graphs

## TL;DR

This paper introduces new centrality measures based on the directional derivatives of Kemeny's constant in undirected graphs, providing tools for analyzing connectivity, sensitivity, and link prediction in network structures.

## Contribution

It defines novel edge and non-edge centrality measures using derivatives of Kemeny's constant, with explicit formulas and algorithms, extending previous work and applicable to various network analyses.

## Key findings

- Derived explicit formulas involving the inverse Laplacian for centrality measures.
- Connected the new measures to existing ones in prior literature.
- Applied the measures to road networks and link prediction tasks.

## Abstract

Kemeny's constant quantifies a graph's connectivity by measuring the average time for a random walker to reach any other vertex. We introduce two concepts of the directional derivative of Kemeny's constant with respect to an edge and use them to define centrality measures for edges and non-edges in the graph. Additionally, we present a sensitivity measure of Kemeny's constant. An explicit expression for these quantities involving the inverse of the modified graph Laplacian is provided, which is valid even for cut-edges. These measures are connected to the one introduced in [Altafini et al., SIMAX 2023], and algorithms for their computation are included. The benefits of these measures are discussed, along with applications to road networks and link prediction analysis. For one-path graphs, an explicit expression for these measures is given in terms of the edge weights.