Information Technology Reference
In-Depth Information
Fig. 21.3
The graph of the
sigmoid function
f
The graph of the sigmoid function
f
isshowninFig.3.
The diffusive component of the network dynamics results from the consensual
interaction between each node
x
i
and the remaining
n
−
1 nodes
x
j
=
i
in the network.
The aggregated effect of these
n
1 interactions is represented as a single consen-
sual interaction between node
x
i
and a virtual node containing a particular weighted
average of the remaining opinion values.
The individual dissensus cost is given by
−
)=
j
=
i
V
(
i
,
j
)
/
(
n
−
1
)
V
(
i
(21.6)
where
2
V
(
i
,
j
)=
f
((
x
i
−
x
j
)
)
(21.7)
and the individual opinion changing cost is
2
U
(
i
)=
f
((
x
i
−
s
j
)
)
.
(21.8)
Summing over the various experts we obtain the collective dissensus cost
V
and
inertial cost
U
1
4
1
2
∑
i
∑
i
V
=
V
(
i
)
U
=
U
(
i
)
(21.9)
/
/
where 1
4and1
2 are conventional multiplicative factors. The full cost function W
is then
W
=(
1
−
λ
)
V
+
λ
U
0
≤
λ
≤
1
.
(21.10)
The consensual network dynamics, which can be regarded as an unsupervised learn-
ing algorithm, acts on the individual opinion variables through the iterative process
x
i
−
ε
∂
W
∂
x
i
=
x
i
−→
x
i
.
(21.11)
The dissensus cost
V
induces a non-linear process of diffusion based on the gradient
term
∂
V
x
i
=
v
i
(
x
i
−
x
i
)
(21.12)
∂
