Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian

Lee DeVille

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.

Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian

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.

Paper Structure

This paper contains 17 sections, 7 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Summary of results of this paper
  4. The model
  5. General model definition
  6. Connection to network consensus models
  7. Interpretation of our model "in coordinates"
  8. Theory
  9. Gradient flow formulation
  10. Linearization around solutions
  11. Energy, convexity, and stability
  12. Existence, or lack thereof, of attracting fixed points
  13. Examples
  14. Twist-like solutions and multistability
  15. Maximal dimensional flows
  16. ...and 2 more sections

Key Result

Proposition 2.6

Let $G=(V,E)$ be the graph induced by eq:K above, i.e. $\{i,j\}\in E \iff \gamma_{ij}\neq 0$. Let $B_1$ be the incidence matrix of $G$, $W_1$ the $\left|{E}\right| \times \left|{E}\right|$ matrix with $(W_1)_{ee} = \gamma_e$, and for now assume that $W_0$ is the $n\times n$ identity matrix. Let $\om and thus fits into the hierarchy eq:nl at the vertex flow level. More generally, if we consider a g

Figures (2)

  • Figure 1: A piece of the simplicial complex triangulating a torus, with vertices, edges, and triangles labeled. We use the convention that the edges are blue(ish), the "left" triangles red, and the "right" triangles green --- and the labels match this coloring scheme as well. Recall, as described in the text, that in the full grid (not pictured) the left/right and up/down edges are identified.
  • Figure 2: Homological solution of the edge flow for the torus triangulation with $m=n=10$ and with the choice $\theta_{0,0,D} = 0, \theta_{0,1,D} = 1$ (recall that there are two degrees of freedom that need to be set).

Theorems & Definitions (21)

  • Definition 2.1: Simplicial complex
  • Definition 2.2: Ordered and weighted complexes
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more