1. Network growth under the constraint of synchronization stability Xingang Wang Department of Physics, ZJU Oct. 17, 2010, Suzhou

2. Network is growing External demand: techonological and information networks, etc. Driving forces Function requirement: social, economy and biology networks, etc. 1. Growth 2. Preferential attachement Smooth growth ? BA growth model (SFN) [1] [1] R. Albert and A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002).

3. Network growth in real world Intermitent growth Uneven growth Ecological networks [3] Other examples: WWW, Internet, authorship, etc. [2] P. Holme and B.J. Kim, Phys. Rev. E 65, 066109 (2002). [3] J.I. Perotti, et.al., Phys. Rev. Lett. 103. 108701 (2009). Technological networks (power-grid) [2]

4. Dynamic growth !

5. Outline: 1.Phenomenon 2.Properties 3.Mechanisms 4.Consequences

6. The model 1.Growth (BA) 2.Preferential attachement (BA) 3.Synchronization stability (functionality) Synchronizable Non-synchronizable [4] A.E. Motter, et.al., EPL 69, 334 (2005). The viewpoint from evolutionary network Node dynamics Growth dynamics Structure

7. Time-scale separation time unit for node addition charactering time for system dynamics (synchronization) No contraint, BA SFN Adiabatic growth, constraint activated Entangled dynamics Master stability function (MSF) [5] [5] M. Barahona and L.M. Pecora, PRL 89, 054101 (2002). synchronizability Eigen-spectrum of Necessary condition: MSF of logistic map a=4

8. A schematic plot on network growth [6] A. Arenas, et.al., Phys. Rep. 469, 93 (2008). ?

9. Questions: 1.Accepting probability 2.Where the new node is connected to 3.The properties of the generated network

10. The boundary eigenvalues Parameters: BA SFN R=4, Constrained BA SFN R=4, Constrained (a) (b)

11. Accepting probability (missing) M  the number of trying additions Intermittent, non-smooth growth P(M)

12. Where missing occurs ?

13. Emergence of super-node

14. Consequence of dynamic growth BA R=4 R=3.8 Super-node SFN (SN-SFN)

15. SN-SFN in practice Internet at AS level [7] Stock market of New York [8] [7] M.E.J. Newman, SIAM Rev. 45, 167 (2002). [8] G. Bonanno, et.al., Phys. Rev. E 68, 046130 (2003). Super-node

16. Topological properties of SN-SFN Network average diameter Averaged clustering coefficient

17. Another question to BA SFN: Is preferential attachement a necessary condition ?

18. Network growth with random attachement Missing distribution Missing location Dynamic growth still

19. Network growth with random attachement Dynamics stabilityPreferential attachement ?

20. SFN with random attachement Star-network Syn. Stability

21. Variation of eigen-spectrum Fast increase Slow increase SN-SFN

22. Direct simulations Local dynamics Missing distribution Super-node

23. Remarks & discussions 1.The use of synchronization constraint 2.The value of R 3.New viewpoint for network evolution 4.Specific form of growth dynamics 5.Long-time evolution 6.Dynamical basis for PA

24. Summary 1. Network growth 2. Preferential attachement Dynamic Growth dynamics

25. Thank you