Network growth under the constraint of synchronization stability

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

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

3. Technological networks (power-grid)[2] Uneven growth Ecological networks [3] [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). Network growth in real world Intermitent growth Other examples: WWW, Internet, authorship, etc.

4. Dynamic growth !

5. Outline: • Phenomenon • Properties • Mechanisms • Consequences

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

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

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

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

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

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

12. Where missing occurs ?

13. Emergence of super-node

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

15. Super-node 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).

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

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

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

19. Dynamics stability Preferential attachement Network growth with random attachement ?

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

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

22. Direct simulations Local dynamics Missing distribution Super-node

23. Remarks & discussions • The use of synchronization constraint • The value of R • New viewpoint for network evolution • Specific form of growth dynamics • Long-time evolution • Dynamical basis for PA

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

25. Thank you