Limiting Moments of the Eigenvalue Distribution of the Watts-Strogatz Random Graph

Undergraduate Honors Thesis

Citations

Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/cz30pt78x

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 to infinity of the eigenvalue distribution. In addition, we conclude by discussing some conjectures about higher moments.

Creator