A Spectral Approach to Kemeny's Constant

TL;DR

This work studies Kemeny’s constant $K(G)$ for graphs through spectral methods and sparsification. It derives a novel eigenvector-based formula for slightly regular networks, linking $K(G)$ to the Laplacian and degree structure via $K(G)={ m tr}(L_G^#D)- ildeeta^ ext{ aisebox{0.5pt}{$ ightarrow$}} rac{||{f k}||^2-rac{{ m vol}(G)^2}{n}}{ ext{vol}(G)}$ when ${f w}$ is an eigenvector, and provides bounds in the general case. The authors then exploit Spielman’s ε-approximation sparsification to define three estimators $K'(G)$, $K''(G)$, and $K'''(G)=K(G')$, with provable error bounds and strong empirical performance on diverse graph families. Finally, they use eigenvalue interlacing to bound $K(G)$ under subgraph operations, yielding practical, tight results for several canonical graphs and illustrating how graph structure governs mixing efficiency in Markov processes.

Abstract

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, providing insight into the global behavior of a Markov chain. We present a novel eigenvector-based formula for computing Kemeny's constant. Moreover, we analyze the impact of network structure on Kemeny's constant. In particular, we use various spectral techniques, such as spectral sparsification of graphs and eigenvalue interlacing, and show that they are particularly useful in this context for deriving approximations and sharp bounds for Kemeny's constant

Theorems & Definitions (20)

• Proposition 3.1
• Proposition 3.2
• Example 3.3
• Example 3.4
• Lemma 4.1: S2017
• Theorem 4.2
• Proposition 4.3
• Theorem 4.4
• Proposition 4.5
• Theorem 5.1