# Polynomial control on stability, inversion and powers of matrices on simple graphs

## TL;DR

This paper introduces new Banach algebras for matrices on simple graphs with polynomial decay, establishing stability, inversion, and power estimates crucial for analyzing large spatial networks.

## Contribution

It develops Beurling-type Banach algebras for graph-based matrices, proves their stability and norm-controlled inversion, and provides polynomial estimates for matrix powers relevant to network analysis.

## Key findings

- Establishes equivalence of $ ext{ell}^p$-stability across different p-values.
- Proves matrices in Beurling subalgebras have norm-controlled inversion.
- Provides polynomial estimates for matrix powers applicable to Markov chains.

## Abstract

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel % Banach algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of ${\mathcal B}(\ell^p), 1\le p\le \infty$, the space of all bounded operators on the space $\ell^p$ of all $p$-summable sequences. The $\ell^p$-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among $\ell^p$-stabilities of matrices in Beurling algebras for different exponents $1\le p\le \infty$, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of ${\mathcal B}(\ell^2)$ have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.