# On The Modified Newman-Watts Small World and Its Random Walk

## TL;DR

This paper investigates a modified Newman-Watts small world model by adding specific long edges to a lattice torus, demonstrating that both the diameter and random walk mixing time grow polynomially with the logarithm of the network size.

## Contribution

It introduces a modification to the Newman-Watts model focusing on long edges proportional to the graph's diameter, analyzing its effects on diameter and mixing time.

## Key findings

- Diameter grows polynomially with n
- Mixing time of random walk grows polynomially with n
- Long edges significantly influence small world properties

## Abstract

It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.