Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
Dynamical Systems
2024-06-19 v2 Disordered Systems and Neural Networks
Adaptation and Self-Organizing Systems
Abstract
We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension, so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional, and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogues of structures seen in related network models.
Cite
@article{arxiv.2010.07421,
title = {Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian},
author = {Lee DeVille},
journal= {arXiv preprint arXiv:2010.07421},
year = {2024}
}