CS 4850 Lecture  Feb 2, 2009
Justin Prieto, jp393; Sujith Vidanapathirana, srv3
Increasing property:
Q
is an increasing property of a random graph
G
if when
p
1
< p
2
,
G
(
n, p
2
) almost
surely has
Q
if
G
(
n, p
1
) almost surely has
Q
. i.e., if we increase the probability of an edge, the property
Q
becomes more likely.
Theorem:
Every increasing property of
N
(
n, p
) has a threshold. (The combinatorial structure does not
actually matter.)
Proof:
To show that
p
(
n
) is a threshold for property
Q
, we need to show that the probability of property
Q
goes from 0 to 1 within a range that is bounded by a multiplicative constant.
We have to show that functions
p
( ) and
p
(1

) are asymptotically equivalent. That is, we need to show
that there exists as constant
m
such that
p
(1

)
≤
mp
( ).
Notational note:
Let
p
(
n,
) be the function
p
(
n
) such that the probability of
Q
is
.
We will also
sometimes write
N
p
for
N
(
n, p
).
Start of Proof
Let 0
<
<
1
2
and let
m
be an integer such that (1

)
m
≤
. We now show that
p
(1

)
≤
mp
( ).
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 Spring '09
 Probability, Probability theory, 2k, Sujith Vidanapathirana, Justin Prieto

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