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Dynamical connectivity matrices reordered by communities for the attractive-neural coupling
B.3 at t
=
=
10 (on the right) are depicted at Fig.7. In case B.3 one
can see (also cf. Fig.8) that number of connections with the attractive coupling is growing in
time, while the strength of the repulsive connections is decreasing, which finally results in the
global synchronization. For the scenario B.2 there is a dynamical balance between attractive
and repulsive coupling with small fluctuations around the mean (Fig.8). Note that even the
averaged strength of the repulsive connections is less than the attractive coupling, the system
dynamics shows a quasi-chaotic behavior.
Fig.9 shows the adjacency matrix for Zachary karate club (red circles), detected communities
by pink squares, predicted links are shown by blue dots. As expected, the dynamical methods
for links prediction tend to make more connections within the established communities first,
followed by merging communities and creating highly overlapped partitions at the higher
hierarchical levels (the upper part at Fig.9). In case of Katz predictor (32), by increasing the
dumping parameter
1 (on the left) and t
we take into account the larger number of paths connecting nodes in the
graph, which in turn results into the larger number of suggested links above a fixed threshold.
Following the concept of dynamical connectivity matrix (20), the process of growing number
of links may be seen as the hierarchical community formation predicted by (32) at different
values of
β
. This process is illustrated at Fig.9, the bottom part. Note that in case of Katz
predictor, the connected graph is also approaching the fully connected graph, but the network
evolution may take a different trajectory compared to the coupled dynamical systems.
β
In
particular, at small values of t and
, the network evolution is similar for both cases (cf.
Fig.9(b) and Fig.9(e)), but with the time the evolution trajectories may follow different paths
(cf. Fig.9(c) and Fig.9(f)), which in turn results in different predictions.
Note that in all cases of the network evolution, we may prioritize the recommended
links based on the soft communities detection (Katz predictor) or the threshold
β
η
(coupled
dynamical systems). We address this issue below in Section 7.
Fig. 5. Karate club: number od communities at different resolution levels.
7. Applications for real wold mobile data
7.1 Community detection in Nokia mobile datasets
To analyze mobile users behavior and study underlying social structure, Nokia Research
Center/Lausanne organized mobile data collection campaign at EPFL university campus
(Kiukkonen et al, 2010). Rich-content datasets (including data from mobile sensors, call-logs,
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