Spectral comparisons between networks with different conductance functions

TL;DR

This paper compares the spectral properties of networks with different conductance functions, analyzing how replacing edge weights affects energy spaces, spectra, and effective resistance, providing new insights into network analysis.

Contribution

It introduces a spectral comparison framework for networks with varying conductance functions, linking energy spaces, spectra, and effective resistance estimates.

Findings

Spectral differences depend on conductance function changes.

Derived bounds for effective resistance between networks.

Computed a spectral invariant for energy space embeddings.

Abstract

For a network consisting of a graph with edge weights prescribed by a given conductance function $c$, we consider the effects of replacing these weights with a new function $b$ that satisfies $b \leq c$ on each edge. In particular, we compare the corresponding energy spaces and the spectra of the Laplace operators acting on these spaces. We use these results to derive estimates for effective resistance on the two networks, and to compute a spectral invariant for the canonical embedding of one energy space into the other.