Foundations of Data Science

For x to be equal to zero, it must differ from its expected value by at least its expected
value. Thus,
Prob(x = 0) ≤ Prob ( |x − E(x)| ≥ E(x) ) .
By Chebychev inequality,
Prob(x = 0) ≤ Var(x)
E 2 (x) ≤ d3 /6 + o(1)
d 6 /36
≤ 6 + o(1). (4.1)
d3 Thus, for d >
3√ 6 ∼ = 1.8, Prob(x = 0) < 1 and G(n, p) has a triangle with nonzero
probability. For d < 3√ 6 and very close to zero, there simply are not enough edges in the
graph for there to be a triangle.
4.2 Phase Transitions
Many properties of random graphs undergo structural changes as the edge probability
passes some threshold value. This phenomenon is similar to the abrupt phase transitions in
physics, as the temperature or pressure increases. Some examples of this are the abrupt
appearance of cycles in G(n, p) when p reaches 1/n and the disappearance of isolated
vertices when p reaches log n . The most important of these transitions is the emergence of
n
a giant component, a connected component of size Θ(n), which happens at d = 1. Recall
Figure 4.1.
Probability Transition
p = 0 Isolated vertices
p = o( 1 ) n
Forest of trees, no component
of size greater than O(log n)
p = d , d < 1
n
All components of size O(log n)
p = d , d = 1 n Components of size O(n 2 3 )
p = d , d > 1 Giant component plus O(log n) components
n
p = √ √
2 Diameter two
p = 1 ln n
2 n
p = ln n
n
p = 1 2
ln n
n
Giant component plus isolated vertices
Disappearance of isolated vertices
Appearance of Hamilton circuit
Diameter O(ln n)
Clique of size (2 − ɛ) log n
Table 1: Phase transitions
For these and many other properties of random graphs, a threshold exists where an
abrupt transition from not having the property to having the property occurs. If there
p
exists a function p (n) such that when lim 1 (n)
= 0, G (n, p
n→∞ p(n)
1 (n)) almost surely does not
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