For G(n,p), the largest rigidity dimension equals the minimum degree below p = C* log n/n, and equals (1/2 + o(1))np above it, up to p = o(n^{-1/2}).
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On the Rigidity of Random Graphs in high-dimensional spaces
For G(n,p), the largest rigidity dimension equals the minimum degree below p = C* log n/n, and equals (1/2 + o(1))np above it, up to p = o(n^{-1/2}).