Foundations of Data Science

Property
cycles 1/n
giant component 1/n
giant component
1 ln n
+ isolated vertices
2 n
connectivity, disappearance ln n
of isolated vertices
n
diameter two
√
2 ln n
n
Threshold
The argument above does not yield a sharp threshold since we argued that E(x) → 0
only under the assumption that p is asymptotically less than 1 . A sharp threshold requires
n
E(x) → 0 for p = d/n, d < 1.
Consider what happens in more detail when p = d/n, d a constant.
n∑
( ) n (k − 1)!
E (x) =
p k
k 2
k=3
= 1 n∑ n(n − 1) · · · (n − k + 1)
(k − 1)! p k
2
k!
k=3
= 1 n∑ n(n − 1) · · · (n − k + 1) d k
2
n k k .
k=3
E (x) converges if d < 1, and diverges if d ≥ 1 . If d < 1, E (x) ≤ 1 2
n∑
equals a constant greater than zero. If d = 1, E (x) = 1 2
n
only the first log n terms of the sum. Since = 1 + i
n−i
≥ 1/2. Thus,
n(n−1)···(n−k+1)
n k
E (x) ≥ 1 2
log ∑n
k=3
Then, in the limit as n goes to infinity
k=3
n−i
n(n−1)···(n−k+1) 1
≥ 1 log ∑n
1
.
n k k 4 k
k=3
log ∑n
1 1
lim E (x) ≥ lim ≥ lim (log log n) = ∞.
n→∞ n→∞ 4 k
k=3
n→∞
n∑
k=3
d k k
and lim E (x)
n→∞
n(n−1)···(n−k+1)
n k 1
k . Consider
≤ e i/n−i , it follows that
For p = d/n, d < 1, E (x) converges to a nonzero constant. For d > 1, E(x) converges
to infinity and a second moment argument shows that graphs will have an unbounded
number of cycles increasing with n.
103