In this paper, we study detection and fast reconstruction of the celebrated Watts-Strogatz (WS) small-world random graph model which aims to describe real-world complex networks that exhibit both high clustering and short average length properties. The WS model with neighborhood size $k$ and rewiring probability probability $\beta$ can be viewed as a continuous interpolation between a deterministic ring lattice graph and the Erdos-Renyi random graph. We study the computational and statistical aspects of detection and recovery of the deterministic ring lattice structure (strong ties) in the presence of random connections (weak ties). The phase diagram in terms of $(k,\beta)$ is shown to consist of several regions according to the difficulty of the problem. We propose distinct methods for these regions.