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

arXiv (Cornell University)

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.

Authors 4

1. Francesco Viganò

   Imperial College London

   Affiliation as printed

   Imperial College London , vigano

2. Tolga Birdal

   Imperial College London

   Affiliation as printed

   Imperial College London ,

3. Michael T. Schaub Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University ,

4. Mauricio Barahona

   Imperial College London

   Affiliation as printed

   Imperial College London ,