1. Dynamics-Based Topology Identification of Complex Networks
Zhigang Zheng
Department of Physics
Beijing Normal University
CCAST, Beijing,

2. Collaborators 杨浦 博士 王群 博士后 Prof. Choi-heng Lai
杨浦 博士
王群 博士后
Prof. Choi-heng Lai
(Temasek Lab, NUS, Singapore)

3. The Hierarchy of Networks
Topology
Dynamics
Function

4. Topology effects on Dynamics
There have been numerous and extensive investigations and reviews
Synchronizations
Spreading, Propagations, transport, ……
Swarming, flocking

5. Recent Reviews:
Complex networks: Structure and dynamics, Phys. Rep. 424, 175 (2006).
Dynamics on Complex Networks and Applications (Special issue), Physica D 224 (2006).
Synchronization in complex networks, Phys. Rep. 469, 93 (2008).

6. Topology VS Dynamics: Scale connections
In fact, the topology effect on collective dynamics is far from being clearly understood.

7. Also In ecology and others……

8. Topology ?
Dynamics
Function

9. The Inverse Problem Black box
How to extract the structural information of network topology from observable data (time series)?
Nonlinearity
Collectiveness
Noise
Node
Dynamics
+Topology
Black box
Node links
(Topology)
Topology??

10. 1. Identification of Microscopic Structures
Complete information of the network topology based on dynamics (adjacent matrix)
Approaches:
Adaptive feedback
Perturbation and response dynamics
Phase dynamics

11. 1.1 Adaptive feedback approach:
References:
D. Yu, M. Righero, and L. Kocarev, PRL 97, (2006).
X. Wu, Physica A 387, 997 (2008).
F. Sorrentino and E. Ott, PRE 79, (2009).
Z.Wu and X. Fu, CPL 26, (2009).
L.Chen, J. Lu, and C.K. Tse, IEEE TRANS. CIR. & SYS. II 56, 310 (2009).

12. D. Yu, M. Righero, and L. Kocarev,
PRL 97, (2006).

13. C(i,j)
X(t)
D(i,j,t)
Y(t)
drive
D(i,j,t)C(i,j)
Y(t)X(t)

14. Shortcomings:
Depending on the node dynamics and knowledge of interactions
Failure in the presence of synchronization (partial or global)

15. 1.2 Perturbation and Response
Dynamics Approach:
References:
M. Timme, PRL 98, (2007).
M. Timme, EPL 76, 367 (2006).
D. Yu, L. Fortuna, and F. Liu, Chaos 18, (2008).
D. Yu and U. Parlitz, EPL 81, (2008).

17. Kuramoto with external input I:
Phase-locked state:
Linear response
Difference
Taylor expansion

18. Introducing
Matrix relation
Where
Final relation
inverse

19. 1.3 Phase Dynamics Approach:
References:
A.Bahraminasab, F. Ghasemi, A.Stefanovska, P.V.E.McClintock, and H.Kantz, PRL 100, (2008).
D.A. Smirnov and B.P. Bezruchko, PRE 79, (2009).

21. 2. Identification of Mesoscopic Structures
Exact exploration of some topology properties based on Dynamics (motif, modules, community, et al.)
References:
S. Boccaletti, M. Ivanchenko, V. Latora, A. Pluchino, and A. Rapisarda, PRE 75, (R) (2007).
Y. Hu, M. Li, P. Zhang, Y. Fan, and Z. Di, PRE 78, (2008).
V. Gudkov, V. Montealegre, S. Nussinov, and Z. Nussinov, PRE 78, (2008).

23. 3. Identification of Macroscopic Structures
A statistical estimation of network topology (A Complex-Network Approach)
References:
S. Bu and I. Jiang, EPL 82, (2008).

25. Basic idea:
The coupling destruction and deformation of the attractor sets
The information about the dominant nodes and degree distribution can be obtained by quantitatively characterizing deformations.

26. Deformation measure

27. The relation between the deformation measure and the node degree

28. Results:

29. Shortcomings:
Time-consuming
Not applicable for all chaotic dynamics
Nonlinear dependence of M on k
Failure for non-chaotic data
Failure for synchronous data

30. Our Recent Works:
Identification of network topology in the presence of synchronization
Estimating statistical properties of network topology (under progress)

31. Dynamical Network
adaptive-feedback
Identifier
Lipschitz condition

32. Evolution of the Error
Lyapunov function
One can prove

33. In the long-term limit:
Linear independence condition
In the presence of synchronization

35. Example: Lorenz node
S: coupling strength

36. s=0.007:
d(11)=c(11)
d(12)=c(12)
d(13)=c(13)
d(14)=c(14)
d(22)=c(22)
d(23)=c(23)
d(24)=c(24)
d(34)=c(34)

37. s=0.11:
d(11)=c(11)
d(12)=c(12)
d(13)=c(13) – 9.92
d(14)=c(14)
d(22)=c(22) – 0.59
d(23)=c(23) – 0.16
d(24)=c(24) – 0.01
d(34)=c(34) – 0.02

38. Synchronization is the obstacle to identification of network topology
Question:
How to identify the network topology in the presence of synchronization?
Most Important: asynchronous information is important
Further—Ways to desynchronize……

39. Basic idea
All information on network topology are embedded in asynchronous transient dynamics
Problem 1: The transient data may be too short!
Problem 2: One cannot get the asynchronous data!

40. Solution 1: The asynchronous data (the transient segment) can be used as a repeat (periodic) drive
Solution 2: One can perturb the system via an external short-term stimuli (e.g., a shock) to drive the system tentatively away from synchrony and to get an asynchronous segment

41. X(t,w)
Y(t), D(i,j,t)
D(i,j,t)C(i,j)

44. Numerical results Identification error: Coupling strength=0.05
(no synchronization)
100<t<400
n=3000

45. Numerical results Identification error: Coupling strength=0.3
(partial synchronization)
2<t<12
n=2000
Coupling strength=0.3 (partial synchronization) n=5000

46. Numerical results Identification error: Coupling strength=0.6
(global synchronization)
2<t<12
n=5000

47. Alternative: Noise can enhance the identification!

48. Still going on……
Identification of the network structure based on output data (dynamics) is a very important issue in both applications and theoretical understanding
Topology estimations in three levels are all crucial (Currently, meso-, macro-, are much more lacking, can we find some dynamics based indices to model topological properties?)
Still far from practical applications

49. Thanks. My permanent address: Department of Physics
Beijing Normal University
Beijing , CHINA
Tel./Fax: