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and as
N
→∞
by simple factoring we obtain
N
k
−
1
z
e
−
z
p
k
=
.
(6.58)
k
N
−
1
−
z
Let us rewrite (
6.58
)as
z
k
k
e
−
z
p
k
=
F
(
N
)
,
(6.59)
!
where the function to be simplified is
(
N
−
1
)
!
(
N
−
1
)
!
F
(
N
)
≡
=
k
.
(6.60)
k
k
(
N
−
1
−
k
)
!
(
N
−
1
−
z
)
(
N
−
1
−
k
)
!
(
N
−
1
)
(
1
−
p
)
Under the condition
p
1, we neglect the
p
-dependence and write
(
−
)
!
(
−
)
!
N
1
N
1
F
(
N
)
≈
=
k
.
(6.61)
(
−
−
)
!
(
−
−
)
k
(
−
−
)
!
(
−
)
N
1
k
N
1
z
N
1
k
N
1
On implementing the lowest order of the Stirling approximation ln
N
!=
N
ln
N
−
N
,
we find, after some algebra, that for
N
→∞
ln
F
(
N
)
=
0
,
(6.62)
thereby yielding
F
(
N
)
≈
1. We conclude that the distribution of
k
links is therefore
given by
z
k
k
e
−
z
p
k
=
,
(6.63)
!
the Poisson distribution.
Note that
z
of (
6.54
) is identical to the mean number of links per node:
z
=
k
.
(6.64)
Let us double check this important property. The mean value
k
is defined by
∞
k
=
kp
k
.
(6.65)
k
=
1
By inserting (
6.59
)into(
6.65
) we obtain
∞
ze
−
z
∞
z
k
−
1
1
z
k
e
−
z
k
=
=
)
!
=
z
.
(6.66)
(
k
−
1
)
!
(
k
−
1
k
=
1
k
=
1
Note the normalization condition
∞
z
k
k
e
−
z
=
1
.
(6.67)
!
k
=
0
Recall our earlier derivation of the Poisson distribution and specifically how to
generalize it beyond the exponential survival probability.
