Stability thresholds for signed Laplacians on locally-connected networks

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}_n². 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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Cong, Lin

Contributors dc:contributor

• DeVille, Lee
• Bronksi, Jared
• Kirkpatrick, Kay
• Rapti, Zoi

• Graph Laplacian
• Dynamics on graphs

Statement dc:rights

• Copyright 2017 Lin Cong

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/97363

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97363