Image Processing Reference
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dynamical connectivity matrix (20) in the form C η (
t
)
to present the evolution of connectivity
in time for a fixed correlation threshold
. Using this approach we consider below several
scenarios of networks evolution with dynamically changing coupling.
η
B.1. Attractive coupling with dynamical updates
As the first step, let's introduce dynamics into static attractive coupling (13).
Using the
dynamical connectivity matrix (20) we may write
sin
,
N
j = 1 F ( η )
˙
θ i (
)= ω i +
(
)
θ j (
) − θ i (
)
t
k c
t
t
t
(21)
ij
describes dynamical attractive coupling, F ( η )
ij
where matrix F ( η ) (
)
(
)=
A ij C η (
) ij
t
t
t
0. Then,
similar to (18), the attractive coupling with a dynamical update may be described as
sin B
,
˙
T
Θ (
t
)= Ω
k c B
(
t
)
D F (
t
)
(
t
)
Θ (
t
)
(22)
(
)
{
a k }
where initial conditions are defined by A ij ; D F
t
is formed from D A with elements
scaled according to C η (
)
t
.
B.2. Combination of attractive and repulsive coupling with dynamical links update
Many biological and social systems show a presence of a competition between conflicting
processes. In case of coupled oscillators it may be modeled as the attractive coupling (driving
oscillators into the global synchronization) combined with the repulsive coupling (forcing
system into a chaotic/random behavior). To allow positive and negative interactions we use
instant correlation matrix R
R + (
R (
(
t
)=
t
)+
t
)
, and separate attractive and repulsive parts
j = 1 r ij ( t ) A ij sin θ j ( t ) − θ i ( t ) k c N
j = 1 | r ij ( t ) | A ij sin θ j ( t ) − θ i ( t ) ,
N
˙
θ i (
)= ω i +
t
k c
(23)
where superscripts denote positive and negative correlations 1 .
Note that the total number of links in the network does not change, at a given time instant
each link performs either attractive or repulsive function.
To obtain the Laplacian presentation we define a dynamical connectivity matrix F
(
)
t
as
element-by-element matrix product
F + (
F (
F
(
t
)=
R
(
t
)
A
=
t
)+
t
)
,
(24)
and present dynamic Laplacian as the following
B T
L F (
t
)=
B
(
t
)(
D F + (
t
)+
D F (
t
))
(
t
)
.
(25)
It allows us to write
m = 1 F ij ( t ) sin θ j ( t ) θ i ( t ) k c
m = 1 F ij ( t ) sin θ j ( t ) θ i ( t ) ,
N
N
˙
θ i (
)= ω n
+
t
k c
(26)
1
For presentation clarity we omit here the correlation threshold
η
.
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