Foundations of Data Science

} {{ }
copies of G
If any graph has three or more edges, then the
m-fold replication has three or more edges.
The m-fold
replication H
} {{ }
copies of G
Even if no graph has three or more edges, the
m-fold replication might have three or more edges.
The m-fold
replication H
Figure 4.12: The property that G has three or more edges is an increasing property. Let
H be the m-fold replication of G. If any copy of G has three or more edges, H has three
or more edges. However, H can have three or more edges even if no copy of G has three
or more edges.
Theorem 4.19 Every increasing property Q of G(n, p) has a phase transition at p(n),
where for each n, p(n) is the minimum real number a n for which the probability that
G(n, a n ) has property Q is 1/2.
Proof: Let p 0 (n) be any function such that
p 0 (n)
lim
n→∞ p(n) = 0.
We assert that almost surely G(n, p 0 ) does not have the property Q. Suppose for contradiction,
that this is not true. That is, the probability that G(n, p 0 ) has the property
Q does not converge to zero. By the definition of a limit, there exists ε > 0 for which
the probability that G(n, p 0 ) has property Q is at least ε on an infinite set I of n. Let
m = ⌈(1/ε)⌉. Let G(n, q) be the m-fold replication of G(n, p 0 ). The probability that
G(n, q) does not have Q is at most (1 − ε) m ≤ e −1 ≤ 1/2 for all n ∈ I. For these n, by
(11.4)
Prob(G(n, mp 0 ) has Q) ≥ Prob(G(n, q) has Q) ≥ 1/2.
Since p(n) is the minimum real number a n for which the probability that G(n, a n ) has
property Q is 1/2, it must be that mp 0 (n) ≥ p(n). This implies that p 0(n)
is at least 1/m
p(n)
p
infinitely often, contradicting the hypothesis that lim 0 (n)
= 0.
n→∞ p(n)
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