Higher-order Cheeger inequalities for graphons

TL;DR

The paper develops a graphon analogue of higher-order Cheeger inequalities, linking the $k$-th Laplacian eigenvalue $\lambda_k$ to the $k$-way expansion $h_W(k)$ for connected graphons. By adapting the Rayleigh quotient framework and higher-order construction used in finite graphs, it proves $\frac{\lambda_k}{2} \le h_W(k) \le O(k^{3.5})\sqrt{\lambda_k}$, with a note that existing Markov-operator methods yield a weaker $O(k^{4})$ bound. The approach combines lower-bound via indicator-function subspaces with an intricate upper-bound argument that builds $k$ disjoint, well-separated, smooth approximations and converts them into disjoint measurable sets of controlled expansion. The results extend the discrete higher-order Cheeger theory to the graphon setting, enriching the spectral theory of graph limits and providing continuum-level tools for dense graph analysis.

Abstract

The higher-order Cheeger inequalities were established for graphs by Lee, Oveis Gharan and Trevisan. We prove analogous inequalities for graphons in this article.

Theorems & Definitions (20)

• Theorem 1.1
• proof : Proof of \ref{['thm:HOCI-graphons']}
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof