Root-to-Leaf Path Random Walks, Normalized Hodge Laplacians, and Cheeger Inequalities on Simplicial Complexes

TL;DR

This paper introduces root-to-leaf path random walks on simplicial complexes, linking them to normalized Hodge Laplacians and deriving Cheeger inequalities to understand spectral bounds.

Contribution

It establishes a novel connection between random walks on simplicial complexes and normalized Hodge Laplacians, leading to new spectral bounds and structural insights.

Findings

Induces natural normalization of coboundary operator and Hodge Laplacians

Derives Cheeger inequalities for normalized Hodge spectrum

Combines up- and down-cases for sharper spectral estimates

Abstract

We introduce root-to-leaf path random walks on double covers of graded signed graphs and analyze their behavior in a general setting. Viewing simplicial complexes within this framework, we show that these walks induce the natural normalization of the coboundary operator and of the Hodge Laplacians while preserving the basic structural features of combinatorial Hodge theory. We then derive Cheeger inequalities for the upper side of the normalized Hodge spectrum, identify the coherent structures governing these bounds, and combine the up- and down-cases into sharper estimates.