  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 $\ln n$.
