1. The intimate relationship between networks' structure and dynamics
Synch. vs. Link. :
The intimate relationship between
networks' structure and dynamics
Stefano Boccaletti
Italian Embassy in Israel
CNR-Institute for Complex Systems
Collaborators
Irene Sendina-Nadal Inmaculada Leyva,
Javier Buldú, Juan A. Almendral
Universidad Rey Juan Carlos, Spain
Daiqing Li, Shlomo Havlin
Bar Ilan University 1

2. Few examples of real networks Biological, technological, and social
METABOLIC
CITATIONS
Nodes: chemicals
Links: biochemical reactions
Nodes: published articles
Links: reference to a published article
THE WWW
Nodes: web pages
Links: hyperlinks

3. The first issue: why a same scaling
The first issue: why a same scaling? Biological, technological, and social
METABOLIC
CITATIONS
THE WWW
△ ISI papers;
○ PRD papers
between 1975 and 1994
[Jeon et al., Nature 407, 651 (2000)]
[Redner Eur. Phys. J. B 4, 131 (1998)]
[Albert et al., Nature 401, 130 (1999)]

4. Previous approaches: Dynamics on a network Dynamics of a network
Collective behaviors determined by specific topological properties:
weighted networks, aging, dynamical weights...
Statistical models (growing or not) to generate SF networks: configuration model, preferential attachment, copying vertex 4 4

5. Our approach to SF networks Dynamics drives topology
A fully dynamically driven growing process is able to reshape the connection topology of an initial graph
Scale-free topologies emerge in connection and only in connection with the realization of an entrainment process
Gross&Blasius, J. Roy Soc Int 5, 2008 5 5

6. The model The pristine network G0
n0 non-identical oscillators, each one with an intrinsic frequency ωoi randomly selected from 0.5±0.25 and described by a phase equation given by: G0 3 4 1
adjacency matrix 2
coupling strength for which G0 is not phase-synchronized 6 5 6 6

7. The model The forcing network G(t)
On top of G0, at later times, nodes are added up to n1 following the phase of an external pacemaker of a driving frequency
Each added node forms m links with nodes of G0.
Each added link corresponds to an unidirectional interaction of strength from the added node to the linked node in G0. The network-phase equation is given by:
G(t1)‏ 3 4 1 2
size evolving adjacency matrix
with 6
time evolving degree of the ith node 5 7 7

8. The model The attaching rule
At each attaching time, the m links formed by the added node are directed to those nodes in G0 whose phases verify more closely a specific condition in the phase difference with the added node (fully dynamically attachment criterion)‏ 8 8

9. Frequency and phase entrainment Scenario in the parameter space dp-ωp
No Entrainment
dp small
adding off
adding on
Entrainment
dp large 9 9

10. Frequency and phase entrainment Scenario in the parameter space dp-ωp
Frequency entrainment
Phase entrainment
0.1
0.5
0.9 10 10

11. 02
Frequency and phase entrainment Scenario in the parameter space dp-ωp
time
time
time
p=0.3
p=0.5
p=0.7 11 11

12. 02 Frequency and phase entrainment Phase relationship dependence12 12

13. The rising of a scale-free network Resulting graphs
No entrainment
Entrainment
dp small
dp large 13 13

14. The rising of a scale-free network Time evolution of the cumulative degree distribution
[Sendiña-Nadal et al., Plos ONE (2008), accepted].
As a consequence of the entrainment a power-law is obtained in the dinamically driven growth process
No entrainment
Entrainment 14 14

15. The rising of a scale-free network Cumulative degree distribution in the parameter space03 15 15

16. 03
The rising of a scale-free network Cumulative degree distribution for different initial G0
Before entrainment
After entrainment 16 16

17. Frequency and phase entrainment Phase locking process
No entrainment
Entrainment 17 17

18. Frequency and phase entrainment Phase locking process
No entrainment
Entrainment 18 18

19. The rising of a scale-free network Final degree vs initial frequency
The distribution is almost uniform implying a random shooting
The nodes whose original frequency mismatch is higher are more likely to be linked 19 19

20. How the network acts under the presence of different
Identificating the overlapped structure of moduli
How the network acts under the presence of different
simultaneous stimulous?
Clusters of synchronization
Overlapping : Formation and dynamics of synchronization interfaces
Method for identificating the overlapping structure of the networks moduli through funtion 20

21. Competition entrainment: non modular network
- Initial Erdös-Reiny model
of phase oscillators, Go
- Inner coupling dnet small
- Initially not synchronized 21

22. Two pacemaker of frequencies The strength of the external
Competition entrainment: non modular network ω1 ω2
Two pacemaker of frequencies
ω1, ω2 unidirectionally link
to a half of Go
The strength of the external
links dp, is choosen high
enough as to assure
entrainment. 22

23. ω1 ω2 Competition entrainment: non modular network
The attaching criteria is
purely dynamic
(phase difference
with the pacemaker).
The global degree distribution
of Go becomes scale-free.
(I. Sendiña-Nadal , et. al.
PLoS ONE (2008)) 23

24. ω1 ω2 Competition entrainment: non modular network
The Go inner coupling dnet is strenghened above dp
The external entrainment
compete with the inner network dynamics 24

25. Competition entrainment: non modular networkω1 ω2
dnet
Time 25

26. Competition entrainment: non modular networkTo
ω2-ω1 26

27. Two big communities connected by
Simplified model of modular network
Time ω
ω1- ω2
Two big communities connected by
a small cluster ω1 ω2 ω3 27

28. Oscillates around ω3 with frequency ωΔ
Simplified model: analitical solution :
Main
communities
Overlapping
community
Oscillates around ω3
with frequency ωΔ 28

29. Even if the overlapping community
Conclusions : detecting overlapping comunities
A new practical way to detect funtional overlapping communities in modular
networks, based on dynamical response
- Detection based on dynamical distance to mean global frecuency
Even if the overlapping community
has only a single node !!
- Useful in modular networks where coordination of communities in the performance of the ensemble is important 29

30. 04 Conclusions and advertisements:
Structure and function of a network are intimately related
An entrainment process is associated with the emergence of a scale-free degree distribution in the graph connectivity.
Coordination of multiple tasks implies the presence of an overlapped structure of moduli T
To know more:
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang,
“Complex Neworks: Structure and Dynamics”
Physics Reports, 424, (2006)
S. Boccaletti
“The synchronized dynamics of complex systems”
Elsevier (2008) 30 30