Approximating Diffusion on Finite Multi-Topology Systems Using Ultrametrics

TL;DR

This paper introduces a lossless way to represent multiple topologies on finite sets using ultrametrics, and applies this to optimize processes like heat flow simulation on city models with spectral analysis of p-adic Laplacians.

Contribution

It proposes a novel ultrametric-based index structure for multi-topology data and analyzes spectral properties of p-adic Laplacians for heat equation approximations.

Findings

Spectra of p-adic Laplacians on finite graphs are characterized.

Error bounds for heat equation solutions using finite operator approximations are provided.

Ultrametrics serve as effective indices for multi-topology data in simulation contexts.

Abstract

Motivated by multi-topology building and city model data, first a lossless representation of multiple $T_0$-topologies on a given finite set by a vertex-edge-weighted graph is given, and the subdominant ultrametric of the associated weighted graph distance matrix is proposed as an index structure for these data. This is applied in a heuristic parallel topological sort algorithm for edge-weighted directed acyclic graphs. Such structured data are of interest in simulation of processes like heat flows on building or city models on distributed processors. With this in view, the bulk of this article calculates the spectra of certain unbounded self-adjoint $p$-adic Laplacian operators on the $L^2$-spaces of a compact open subdomain of the $p$-adic number field associated with a finite graph $G$ with respect to the restricted Haar measure. as well as to a Radon measure coming from an ultrametric on the vertices of $G$ with the help of $p$-adic polynomial interpolation. In the end, error bounds are given for the solutions of the corresponding heat equations by finite approximations of such operators.