The Eigenvalue Distribution of the WattStrogatz Random Graph
Abstract
This paper studies the eigenvalue distribution of the WattsStrogatz random graph, which is known as the "smallworld" random graph. The construction of the smallworld 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 selfloops 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 smallworld random graph. We compute the first moment, second moment, and prove the limiting third moment as n goes to infinity of the eigenvalue distribution.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.00332
 arXiv:
 arXiv:2009.00332
 Bibcode:
 2020arXiv200900332N
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics
 EPrint:
 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