Mathematics and Computer Science II pp 147-159 | Cite as

# The Diameter of a Long-Range Percolation Graph

## Abstract

e 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∥ > 1, and with probability 1 if ∥x - y∥ = 1, for some parameters β, s > 0. This model was introduced by Benjamini and Berger [2], who obtained bounds on the diameter of this graph for the one-dimensional case 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 < η_{1} < η_{2} < 1, it is at most N^{η2 }when s = 2d, and is at least N^{η2 } when d = 1, s = 2, β < 1 or when s > 2d. We also provide a simple proof that the diameter is at most log^{0(1)} N with high probability, when d < s < 2d, established previously in [2].

## Keywords

Short Path Random Graph Side Length Percolation Model Circle Graph## Preview

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## References

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