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Unformatted text preview: tical (µ > 1)
Subcritical: µ < 1
Since E[Zn ] = µn , we have
�∞
�
��
∞
1
E[Z ] = E ∑ Zn = ∑ E Zn =
< ∞,
1−µ
n =1
n =1
(some care is needed in the second equality).
This implies that Z < ∞ with probability 1 and P(extinction ) = 1.
7 Networks: Lecture 3 Introduction Branching Processes (Continued)
Supercritical: µ > 1
Recall p0 = P(ξ = 0). If p0 = 0, then P(extinction ) = 0.
Assume p0 > 0.
We have ρ = P(extinction ) ≥ P(Z1 = 0) = p0 > 0. We can write the following ﬁxedpoint equation for ρ:
ρ= ∞ ∑ pk ρ k = E [ ρ ξ ] ≡ Φ ( ρ ) . k =0 We have Φ(0) = p0 (using convention 00 = 1) and Φ(1) = 1
Φ is a convex function (Φ (ρ) ≥ 0 for all ρ ∈ [0, 1]), and Φ (1) = µ > 1.
1 Φ ( ρ* )= ρ* 0.8 µ 0.6 0.4 0.2 p0 0 0 ρ* 0.2 0.4 ρ 0.6 0.8 1 Figure: The generating function Φ has a unique ﬁxed point ρ∗ ∈ [0, 1). 8 Networks: Lecture 3 Introduction Phase Transitions for Erd¨sRenyi Model
o
Erd¨sRenyi model is completely speciﬁed by the link formation probability
o
p (n ).
For a given property A (e.g. connectivity), we deﬁne a threshold function
t (n ) as a function that satisﬁes:
P(property A) → 0
P(property A) → 1 p (n )
→ 0, and
t (n ) if
if p (n )
→ ∞.
t (n ) This deﬁnition makes sense for “monotone or increasing properties,”
i.e., properties such that if a given network satisﬁes it, any
supernetwork (in the sense of set inclusion) satisﬁes it.
When such a threshold function exists, we say that a phase transition occurs
at that threshold.
Exhibiting such phase transitions was one of the main contributions of the
seminal work of Erd¨s and Renyi 1959.
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9 Networks: Lecture 3 Introduction Phase Transition Example
Deﬁne property A as A = {number of edges > 0}. We are looking for a threshold for the emergence of the ﬁrst edge.
Recall E[number of edges] =
p (n ) n (n −1)
p (n )
2 ≈ n2 p (n ).
2 Assume 2/n2 → 0 as n → ∞. Then, E[number of edges]→ 0, which implies
that P(number of...
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 Fall '09
 Acemoglu

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