Foundations of Data Science

1
Prob(x > 0)
0
1 1
n 1+ɛ n log n
1
n
log n
n
1
2
0.6
n
0.8
n
1
n
1.2
n
1.4
n
1−o(1)
n
1
n
1+o(1)
n
(a) (b) (c)
Figure 4.5: Figure 4.5(a) shows a phase transition at p = 1 . The dotted line shows an
n
abrupt transition in Prob(x) from 0 to 1. For any function asymptotically less than 1 , n
Prob(x)>0 is zero and for any function asymptotically greater than 1 , Prob(x)>0 is one.
n
Figure 4.5(b) expands the scale and shows a less abrupt change in probability unless
the phase transition is sharp as illustrated by the dotted line. Figure 4.5(c) is a further
expansion and the sharp transition is now more smooth.
p
have the property, and when lim 2 (n)
= ∞, G (n, p
n→∞ p(n)
2 (n)) almost surely has the property,
then we say that a phase transition occurs, and p (n) is the threshold. Recall that G(n, p)
“almost surely does not have the property” means that the probability that it has the
property goes to zero in the limit, as n goes to infinity. We shall soon see that every
increasing property has a threshold. This is true not only for increasing properties of
G (n, p), but for increasing properties of any combinatorial structure. If for cp (n), c < 1,
the graph almost surely does not have the property and for cp (n) , c > 1, the graph
almost surely has the property, then p (n) is a sharp threshold. The existence of a giant
component has a sharp threshold at 1/n. We will prove this later.
In establishing phase transitions, we often use a variable x(n) to denote the number
of occurrences of an item in a random graph. If the expected value of x(n) goes to zero as
n goes to infinity, then a graph picked at random almost surely has no occurrence of the
item. This follows from Markov’s inequality. Since x is a nonnegative random variable
Prob(x ≥ a) ≤ 1 E(x), which implies that the probability of x(n) ≥ 1 is at most E(x(n)).
a
That is, if the expected number of occurrences of an item in a graph goes to zero, the
probability that there are one or more occurrences of the item in a randomly selected
graph goes to zero. This is called the first moment method.
The previous section showed that the property of having a triangle has a threshold at
p(n) = 1/n. If the edge probability p 1 (n) is o(1/n), then the expected number of triangles
goes to zero and by the first moment method, the graph almost surely has no triangle.
However, if the edge probability p 2 (n) satisfies p 2(n)
→ ∞, then from (4.1), the probability
1/n
of having no triangle is at most 6/d 3 +o(1) = 6/(np 2 (n)) 3 +o(1), which goes to zero. This
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