Information Technology Reference
In-Depth Information
where
n
is the number of users in the social network. We can have the next state of
users as follows:
0
1
a
11
a
12
a
1
n
.
.
@
A
:
.
.
a
21
vA
¼ð
v
1
;
v
2
; ...;
v
n
Þ
(3.6)
.
.
.
.
.
a
n
1
...
...
a
nn
Hence, we can obtain
m
-time steps later,
0
@
1
A
m
a
11
a
12
a
1
n
.
.
.
.
a
21
vA
m
¼ð
v
1
;
v
2
; ...;
v
n
Þ
(3.7)
.
.
.
.
.
a
n
1
...
...
a
nn
as the result of the Markov chain model. We use the logical sum instead of the
algebraic sum in our simulation for the memory effect.
3.5 Analysis
3.5.1 Logistic Curves
Rogers (
2003
) has investigated a wide variety of diffusion phenomena. Figure
3.1
shows logistic curves
1
f
a
ð
t
Þ¼
(3.8)
e
at
1
þ
introduced by Verhulst (
1838
) with different continuous parameter
a
[0, 2]. When
f
a
(
t
) means the probability to reach a particular information at time
t
, the function
(
3.8
) describes the percolation of the information.
In fact,
f
a
(
t
) satisfies the ordinary differential equation
∈
d
dt
f
a
ð
t
Þ¼
af
a
ð
t
Þð
1
f
a
ð
t
ÞÞ:
(3.9)
This means the increase rate or slope of
f
a
(
t
) is proportional both to
f
a
(
t
) and
1
f
a
(
t
). It is clear that
f
a
(
t
) becomes flat far from the origin, where
f
a
(
t
) takes the
value near 0 or 1. It also should be remarked that
f
a
(
t
) increases the most rapidly at
t
¼
0, where
f
a
(
t
) takes the value of 1
/
2. We know the quadratic form
x
(1
x
)
takes the value between 0 and 1
/
4.
