what-when-how
In Depth Tutorials and Information
innovation can be simply described as two problems: When will one user receive
the innovation? What's the probability that the user will adopt the innovation?
To solve this problem, Song et al. [73] proposed a rate-based information low
model based on the foundation of Continuous-Time Markov Chain (CTMC). he
model can identify where information should low to, and who will most quickly
receive the information.
he definition of CTMC is
Deinition1
: A Continuous-Time Markov Chain is a continuous-time stochas-
tic process
{
t
ʺ
0
s.t.
∀ ≥
X t
( ),
}
0
, and
∀
i
s t
,
,
j x h
, ( )
.
+
=
=
=
≤ ≤
P X t
{
(
s
)
j X t
|
( )
i X h
,
( )
x h
( ),
0
h
t
}
(3.20)
=
+
=
=
P X t
{
(
s
)
j X t
|
( )
i
}
A CTMC satisfies the Markov property and takes value from a discrete state space.
Assume that the transition probabilities are independent from the initial time
t
,
which means the chain is time homogeneous and denotes
P
ij
(
s
) as the transition
probability from
i
to
j
over
s
time period.
Deinition2
: Define the transition rate matrix as
…
⎛
q
q
⎞
0 0
,
0 1
,
⎜
⎜
⎟
⎟
=
Q
q
q
(3.21)
1 0
,
11
,
⎝
⎠
where
Δ
P
(
t
)
P X
{
=
j X
|
=
i
}
ij
=
t
+
Δ
t
t
=
≠
(3.22)
q
lim
lim
(
i
j
)
ij
Δ
Δ
t
t
Δ
→
Δ
→
t
0
t
0
as the probability per time unit that the CTMC makes a transition from state
i
to
state
j
or the transition rate. hus, the total transition rate out of state
i
is
…
−
⎛
q
q
⎞
⎛
q
q
⎞
0 0
,
0 1
,
0
0 1
,
⎜
⎜
⎟
⎜
⎜
⎟
⎟
=
⎟
=
−
q
…
Q
q
q
q
(3.23)
1 0
,
11
,
1
,
0
1
⎝
⎠
⎝
⎠
Deinition3
: Define the time until the CTMC makes a transition and leaves
state
i
, given that the CTMC is currently in state
i
, as the state-staying time of the
chain in state
i
,
T
i
.
T
=
inf{ :
t X
≠
i X
|
=
i
}
(3.24)
i
t
0
