  This paper focuses on the problem of modeling for small world effect on
complex networks. Let's consider the supercritical Poisson continuous
percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long
edges (short cuts)" randomly to the largest percolation cluster, we obtain a
random graph $\mathscr G_n$. In the present paper, we first prove that the
diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we
call it the Poisson Geometry Small World. Secondly, we prove that the random
walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the random
walk mixes in time at most polynomially large in $\ln n$.
