Local ultrametric approximation of graph distance based Laplacian diffusion

TL;DR

This paper introduces a heuristic polynomial-time algorithm for approximating graph-distance Laplacian matrices with ultrametric structures, providing error estimates for solutions to related heat equations.

Contribution

It presents a novel method for locally ultrametric approximation of graph Laplacians using Vietoris-Rips graphs, with error analysis for heat equation solutions.

Findings

Algorithm effectively approximates graph-distance Laplacians.

Error bounds are established for heat equation solutions.

Ultrametric approximation improves spectral analysis of graphs.

Abstract

The error estimation for eigenvalues and eigenvectors of a small positive symmetric perturbation on the spectrum of a graph Laplacian is related to Gau{\ss} hypergeometric functions. Based on this, a heuristic polynomial-time algorithm for finding an optimal locally ultrametric approximation of a graph-distance power Laplacian matrix via the Vietoris-Rips graph based on the graph distance function is proposed. In the end, the error in the solution to the graph Laplacian heat equation given by extension to a locally p-adic equation is estimated.