Stability thresholds for signed Laplacians on locally-connected networks
type: CreativeWork
description: In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely $\mathcal{C}_n^2$. In chapter 4, we extend the previous definition and question to $\mathcal{C}_n^{(k)}$, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k>3$. We then give several conjectures about this topic, which are supported by numerical experiments.
abstract: In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely $\mathcal{C}_n^2$. In chapter 4, we extend the previous definition and question to $\mathcal{C}_n^{(k)}$, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k>3$. We then give several conjectures about this topic, which are supported by numerical experiments.
author: Cong, Lin
language: en
keywords: Graph Laplacian, Dynamics on graphs
date published: 2017-04-19
date modified: 2026-07-22T22:24:34Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/97363