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 Tdn with volume nd. By adding “long edges (short cuts)” randomly to the largest percolation cluster, we obtain a random graph Gn. In the present paper, we first prove that the diameter of Gn grows at most polynomially fast in lnn and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on Gn possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in lnn.