The Eigenvalue Distribution of the Watt-Strogatz Random Graph
Abstract
This paper studies the eigenvalue distribution of the Watts-Strogatz random graph, which is known as the "small-world" random graph. The construction of the small-world random graph starts with a regular ring lattice of n vertices; each has exactly k neighbors with equally k/2 edges on each side. With probability p, each downside neighbor of a particular vertex will rewire independently to a random vertex on the graph without allowing for self-loops or duplication. The rewiring process starts at the first adjacent neighbor of vertex 1 and continues in an orderly fashion to the farthest downside neighbor of vertex n. Each edge must be considered once. This paper focuses on the eigenvalues of the adjacency matrix A_n, used to represent the small-world random graph. We compute the first moment, second moment, and prove the limiting third moment as n goes to infinity of the eigenvalue distribution.
Cite
@article{arxiv.2009.00332,
title = {The Eigenvalue Distribution of the Watt-Strogatz Random Graph},
author = {Poramate Nakkirt},
journal= {arXiv preprint arXiv:2009.00332},
year = {2021}
}
Comments
Added references for section 1 and 3, added footnote at the first page, added acknowledgements section, removed unnecessary figures. This paper is largely based on the author's undergraduate honors thesis, but a thesis version is only used for the consideration for the honors degree, not intended for publication