Image Processing Reference
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partitions
{P
}
with the decreasing numbers of communities.
t
Furthermore,
as
shown
in
(Evans & Lambiotte,
2009),
we
may
define
a
time-varying
modularity Q
(
t
)
by linear terms in time expansion for R
(
t
)
at t
0,
(
) (
) ·
(
)+
·
=
(
)
R
t
1
t
R
0
t
Q
Q
t
,
(10)
which after substitution (6) and (9) gives
A ij
2 m t
4 m 2 .
d i d j
)+
c k ∈P
Q
(
t
)=(
1
t
(11)
i , j
c k
In the following we apply time-dependent modularity maximization (11) using the greedy
search to find hierarchical structures in networks beyond modularity maximization Q max in
(1). This approach is useful in cases where maximization of (1) results in a very fragmental
structure with a large number of communities. Also it allows us to evaluate the stability of
communities at different resolution levels. However, since the adjacency matrix A is not time
dependent, the time-varying modularity (11) can not be used to make predictions beyond the
given topology.
3. Topology detection using coupled dynamical systems
3.1 Laplacian formulation of network dynamics
Let's consider an undirected weighted graph G
= {
}
with N nodes and E edges, where
each node represents a local dynamical system and edges correspond to local coupling.
Dynamics of N locally coupled dynamical systems on the graph G is described by
V , E
j = 1 A ij ψ x j ( t ) x i ( t ) ,
N
x i (
t
)=
q i (
x i (
t
)) +
k c
(12)
where q i (
x i )
describes a local dynamics of state x i ; A ij is a coupling strength between nodes i
ψ ( · )
and j ;
is a coupling function; k c is a global coupling gain.
In case of weakly phase-coupled oscillators the dynamics of local states is described by
Kuramoto model (Acebron et al, 2005; Kuramoto, 1975)
j = 1 A ij sin θ j ( t ) θ i ( t ) .
N
˙
θ i
(
t
)= ω i
+
k c
(13)
( θ ) θ
Linear approximation of coupling function sin
in (13) results in the consensus model
(Olfati-Saber et al, 2007)
j = 1 A ij θ j ( t ) θ i ( t ) ,
N
˙
θ i
(
t
)=
k c
(14)
which for a connectivity graph G may be written as
˙
Θ(
)=
Θ(
)
t
k c L
t
,
(15)
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