The Eigenvalue Distribution of the Watt-Strogatz Random Graph

TL;DR

This paper analyzes the eigenvalue distribution of Watts-Strogatz small-world graphs, deriving moments of the eigenvalues' distribution as the graph size grows infinitely large.

Contribution

It provides the first and second moments and proves the limiting third moment of the eigenvalue distribution for Watts-Strogatz graphs.

Findings

Computed the first and second moments of the eigenvalue distribution.

Proved the limiting third moment as graph size approaches infinity.

Characterized the eigenvalue distribution for large Watts-Strogatz graphs.

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.