Geodesic
Diameters of random distance graphs
Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 24-35 
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/
@article{CMA_2016_100_a3,
     author = {L. Iskhakov and M. Mironov},
     title = {Diameters of random distance graphs},
     journal = {Contemporary Mathematics and Its Applications},
     pages = {24--35},
     year = {2016},
     volume = {100},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMA_2016_100_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.