# Adaptive aggregation on graphs

## TL;DR

This paper extends a posteriori error estimation techniques from finite element methods to general graphs and Hilbert spaces, and uses these estimates to develop adaptive aggregation methods for graph Laplacians.

## Contribution

It introduces a generalized a posteriori error estimator for graphs and Hilbert spaces, and applies it to create adaptive aggregation techniques for graph Laplacians.

## Key findings

- The estimator effectively assesses aggregation quality.
- The reshaping algorithm improves numerical results.
- The approach is validated on multiple numerical examples.

## Abstract

We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error estimators. We use these estimates to construct aggregation based coarse spaces for graph Laplacians. The estimator is used to assess the quality of an aggregation adaptively. Furthermore, a reshaping algorithm based is tested on several numerical examples.