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3.3.2.2 Transition Probability
Estimate the transition probability based on the instances on the interstate transi-
tion and the time delay on each transition. Given the out-state rate, we estimate the
transition probability from user i to user j as
exp(
=
P
q
q t
)
( )
(3.28)
ij
i
i
ij c
c
where t ij (c) is defined as the interstate diffusion time from node i to node j on
c instances.
According to Equation 28, we have
q
=
q P
(
i
j
)
(3.29)
ij
i
ij
which we define as the interpersonal diffusion rate from user i to user j . hus, we
have all the elements in the Q matrix ready for use.
3.3.2.3 Recommendation Algorithm
For the recommendation problem, given at time t = 0, the user i adopts an item,
and then the information starts to low from this user to others in the network. We
predict users' preferences of information by estimating who will most likely adopt
the item by time t = τ ; in other words, information will low to them. To predict
users' preferences by time t = τ , we estimate the probability that the information
lows from user i to others as the probability that transition i j ( j i ) is enabled
in [0, τ ] as L ( j | i , τ ), which is the ( i , j )-th element in L ( τ ), with
τ
= 0
L
( )
τ
P t dt
( )
(3.30)
where P ( t ) is the transition probability matrix with ( i , j )-th entry P ij ( t ).
Formally, when the state space is finite, we can estimate the transition prob-
ability by solving
P t
'( )
=
P t Q
( )
(3.31)
P
( )
0
=
I
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