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Stat206: Random Graphs and Complex Networks
Spring 2003
Lecture 2: Branching Processes
Lecturer: David Aldous
Scribe: Lara Dolecek
Today we will review branching processes, including the results on extinction and survival probabilities
expressed in terms of the mean and the generating function of a random variable whose distribution models
the branching process. In the end we will brieﬂy state some more advanced results.
Introduction
Let’s start by considering a random variable
X
. If possible values of
X
are non negative integers, then for
p
i
=
P
(
X
=
i
), the sequence (
p
i
,i
≥
0) denotes the distribution of
X
. Recall that
E
[
X
] =
X
i
≥
0
ip
i
Some of the standard distributions and their parameters are:
•
Binomial
(
n,p
),
•
Geometric
(
p
),
•
Poisson
(
λ
),
•
Normal
(
μ,σ
2
), and
•
Exponential
(
λ
).
Branching Processes
GaltonWatson Branching Process
In the GaltonWatson Branching Process (GWBP) model the parameter is a probability distribution (
p
0
,p
1
,...
),
taken on by a random variable
ξ
. For the GWBP model we have the following rule.
Branching Rule
: Each individual has a random number of children in the next generation. These random
variables are independent copies of
ξ
and have a distribution (
p
i
).
Let us now review some standard results about branching processes. For simplicity let
μ
denote E[
ξ
]. Let
Z
n
denote the number of individuals in the nth generation. By default, we set
Z
0
= 1, and also exclude the
case when
ξ
is a constant.
Notice that a branching process may either become extinct or survive forever. We are interested under what
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 Spring '07
 DavidAldous

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