Decay bounds for Bernstein functions of Hermitian matrices with applications to the fractional graph Laplacian

TL;DR

This paper derives new decay bounds for Bernstein functions of Hermitian matrices, including fractional powers, using integral representations, with applications to fractional graph Laplacians and nonlocal network dynamics.

Contribution

It introduces novel decay bounds for Bernstein functions of matrices by leveraging exponential decay results and integral representations, improving understanding of fractional graph Laplacians.

Findings

Decay bounds for Bernstein functions of matrices derived

Power law decay in fractional graph Laplacian strengthened

Results applicable to nonlocal network dynamics

Abstract

For many functions of matrices $f(A)$, it is known that their entries exhibit a rapid -- often exponential or even superexponential -- decay away from the sparsity pattern of the matrix $A$. In this paper we specifically focus on the class of Bernstein functions, which contains the fractional powers $A^\alpha$, $\alpha \in (0,1)$ as an important special case, and derive new decay bounds by exploiting known results for the matrix exponential in conjunction with the L\'evy--Khintchine integral representation. As a particular special case, we find a result concerning the power law decay of the strength of connection in nonlocal network dynamics described by the fractional graph Laplacian, which improves upon known results from the literature by doubling the exponent in the power law.