Diameters of random distance graphs
Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 24-35.

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This paper contains an almost exhaustive description of all cases where a distance random graph asymptotically almost surely has diameter 1, 2, or greater than 2.
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     author = {L. Iskhakov and M. Mironov},
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L. Iskhakov; M. Mironov. Diameters of random distance graphs. Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 24-35. http://geodesic.mathdoc.fr/item/CMA_2016_100_a3/