Higher-order interaction model from geometric measurements

TL;DR

This paper introduces a higher-order geometric consensus model based on simplicial complexes, extending the linear model to higher dimensions, and demonstrates convergence to lower-dimensional affine subspaces through mathematical analysis and simulations.

Contribution

It proposes a novel higher-order consensus model using simplicial geometry, generalizing the linear model and analyzing its convergence properties.

Findings

Model converges to an (n-1)-dimensional affine subspace.

The higher-order model generalizes the linear consensus model.

Numerical simulations confirm theoretical convergence results.

Abstract

We introduce a higher simplicial generalization of the linear consensus model which shares several common features. The well-known linear consensus model is a gradient flow with a sum of squares of distances between each pair of points. Our newly suggested model is also represented as a gradient flow equipped with total $n$-dimensional volume functional consisting of $n+1$ points as a potential. In this manner, the linear consensus model coincides with the case of $n=1$ where distance is understood as the 1-dimensional volume. From a simple mathematical analysis, one can easily show that the linear consensus model (a gradient flow with 1-dimensional volume functional) collapses to one single point, which can be considered as a 0-complex. By extending this result, we show that a solution to our model converges to an $(n-1)$-dimensional affine subspace. We also perform several numerical simulations with an efficient algorithm that reduces the computational cost.