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We consider the following long-range percolation model: an undirected graph with the node set {0, 1, ..., N}d, has edges (x, y) selected with probability ≈ β/||x -y||s if ||x - y|| s d = 1 and for various values of s, but left cases s = 1, 2 open. We show that, with high probability, the diameter of this graph is Θ(log N/log log N) when s = d, and, for some constants 0 ' η1 η2 1, it is at most Nη2 when s = 2d, and is at least Nη1 when d = 1, s = 2, β 1 or when s 2d. We also provide a simple proof that the diameter is at most log O(1) N with high probability, when d s 2d, established previously in [2].