Foundations of Data Science

√
ln n
Setting p = c , n
E (x) ∼ = n2
2
( √
1 − c
∼ =
n 2
2 e−c2 ln n
∼ =
1
2 n2−c2 .
ln n
n
) (1 )
− c
2 ln n n
n
√
ln n
For c > √ 2, lim E (x) = 0. Thus, by the first moment method, for p = c with
n→∞ n
c > √ 2, G (n, p) almost surely has no bad pair and hence has diameter at most two.
Next, consider the case c < √ 2 where lim E (x) = ∞. We appeal to a second moment
n→∞
argument to claim that almost surely a graph has a bad pair and thus has diameter greater
than two.
E(x 2 ) = E
( ∑
i