Information Technology Reference
In-Depth Information
where
is the probability that a new element makes a link to the
j
th node, and is
given by the relative frequency form
(
k
j
)
k
j
j
(
k
j
)
=
k
j
.
(6.6)
Thus,
m
is the fraction of new nodes that make a link to the
j
th node; that is, they
are attracted to the
j
th node at time
t
. We replace the sum in (
6.6
)by(
6.4
) to obtain in
place of (
6.5
)
(
k
j
)
dk
j
dt
=
k
j
2
t
,
(6.7)
which has the general solution
m
t
t
j
β
k
j
(
t
)
=
,
(6.8)
where
t
j
is the time at which the
j
th node is established and
β
=
1
/
2
.
(6.9)
From the form of the solution for the increase in the number of connections with time
we see that the older elements have a shorter
t
j
and consequently a larger number of
links than do the younger nodes. In the literature the number of links is called the degree
of the node. Equation (
6.8
) quantifies the rich-get-richer mechanism referred to above.
To determine how the Pareto distribution follows from the solution (
6.8
) we introduce
(
k
j
<
)
P
k
, the probability that the number of connections
k
for a generic node exceeds
k
j
(
)
. An explicit expression for this quantity can be determined by inverting the general
solution to obtain for the time of establishing the
j
th node
t
t
m
k
j
1
/β
t
j
=
(6.10)
and noting that if
k
>
k
j
then
t
m
k
j
1
/β
t
j
>
.
(6.11)
Thus, the probability for the connectivity can be obtained from the probability in terms
of time,
P
t
j
>
1
/β
t
m
k
j
P
(
k
j
<
k
)
=
.
(6.12)
Consequently, the probability that, out of the total number of nodes at time
t
,
k
j
are
connected to the
j
th node is
P
t
j
>
1
/β
1
1
/β
t
m
k
j
m
k
j
t
dt
j
N
t
m
0
+
=
)
=
−
.
(6.13)
(
t
t
1
/β
(
)
t
m
/
k
j
