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or in matrix form
sin B T
sin B T
.
˙
Θ (
t
)= Ω
k c B
(
t
)
D F + (
t
)
(
t
) Θ (
t
)
+
k c B
(
t
)
D F (
t
)
(
t
) Θ (
t
)
(27)
B.3. Combination of attractive and initially neutral coupling with dynamical links update
Negative correlations (resulting in repulsive coupling) are typically assigned between nodes
which are not initially connected. However, in many cases this scenario is not realistic.
For example, in social networks, the absence of communications between people does not
necessary indicate conflicting (negative) relations, but often has a neutral meaning. To take
this observation into account we modified second term in (23) such that it sets neutral initial
conditions to unconnected nodes in adjacency matrix A . In particular, system dynamics with
links update (24) and initially neutral coupling is described by
j = 1 F ij ( t ) sin θ j ( t ) θ i ( t ) + k c N
cos
,
N
˙
j = 1 F ij ( t )
θ i (
t
)= ω i +
k c
θ j (
t
) θ i (
t
)
(28)
or in the matrix form
sin B T
cos B T
.
˙
Θ (
t
)= Ω
k c B
(
t
)
D F + (
t
)
(
t
) Θ (
t
)
k c B
(
t
)
D F (
t
)
(
t
) Θ (
t
)
(29)
Then a dynamical interplay between the given network topology and local interactions drives
the connectivity evolution. We evaluated the scenarios above using different clustering
measures (Manning et al, 2008) and found that scenario B.3 shows the best performance.
In the following we use coupled system dynamics approach to predict networks' evolution
and to make missing links predictions and recommendations. Furthermore, the suggested
approach allows us also to predict repulsive relations in the network based on the network
topology and links dynamics.
4. Overlapping communities
4.1 Multi-membership
In social networks people belong to several overlapping communities depending on their
families, occupations, hobbies, etc. As the result, users (presented by nodes in a graph)
may have different levels of membership in different communities. This fact motivated us
to consider multi-community membership as edge-weights to different communities and
partition edges instead of clustering nodes.
As an example, we can measure a membership g j
of node k in j -th community as a
number of links (or its weight for a weighted graph) between the k -th node and other nodes
within the same community, g j (
(
k
)
k
)=∑ i c j w ki Then, for each node k we assign a vector
(
)=[
(
)
(
)
,..., g N c (
)]
∈{
}
g
k
g 1
k
, g 2
k
k
, k
1,..., N
which presents the node membership (or
{
c 1 ,..., c N c }
(
)
participation) in all detected communities
. In the following we refer g
k
as a
soft community decision for the k -th node.
To illustrate the approach, overlapping communities derived from benchmark karate club
social network (Zachary, 1977) and membership distributions for selected nodes are depicted
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