1. Phase Transitions of Complex Networks and related Xiaosong Chen Institute of Theoretical Physics Chinese Academy of Sciences CPOD-2011, Wuhan

2. Outline  Phase transitions and critical phenomena, universality and scaling universality and scaling  Complex networks and percolation phase transition transition  Phase transitions of two-dimensional lattice networks under a generalized AP process AP process  Conclusions

3. Phase diagram of normal fluids

4. Phase transitions and critical phenomena Phase: homogenous, equilibrium, macroscopic scale. For example: For example: gases, liquids, solids, plasma,…… gases, liquids, solids, plasma,…… Phase transitions: discontinuous: abrupt change of order parameter (first-order) (first-order) continuous: continuous change of order parameter (critical ) (critical ) divergent response functions ← correlation length divergent response functions ← correlation length Gases → plasma : no phase transition Gases → plasma : no phase transition

5. Scaling and universality in critical phenomena Correlation length: ∞  0  t|  t = (T-T c )/T c  Scaling ： f s (t,h) = A 1 t d  W ( A 2 h t  ) f s (t,h) = A 1 t d  W ( A 2 h t  )  Finite-size scaling ： f s (t,h,L) = L - d  Y (t L   h L  ) f s (t,h,L) = L - d  Y (t L   h L  )

6. Universality critical exponents, scaling functions … depend only on (d,n) depend only on (d,n)  d: dimensionality of system  n: number of order parameter components Irrelevant with microscopic details of systems Irrelevant with microscopic details of systems

7. Complex networks consist of:  nodes  nodes  edges  edges examples: random lattice lattice scale free scale free small world…… small world……

8. Percolation phase transition in networks  Begin with “N” isolated nodes  “m” edges are added (different ways) when “m” small: many small clusters when “m” small: many small clusters when “m” large enough: when “m” large enough: size of the largest cluster / N  finite size of the largest cluster / N  finite emergence of a new phase  percolation transition For a review: Rev. Mod. Phys. 80, 1275 (2008)

9. Percolation phase transition of random network (emergence of a giant cluster ) First-order phase transition

10. The Achilioptas process Choosing two unoccupied edges randomly Choosing two unoccupied edges randomly The edge with the minimum product of the cluster sizes is connected. The edge with the minimum product of the cluster sizes is connected.

11. Is the explosive percolation continuous or first-order? Support to be first-order transition: R. M. Ziff, Phys. Rev. Lett. 103, 045701 (2009). R. M. Ziff, Phys. Rev. Lett. 103, 045701 (2009). Y. S. Cho et al., Phys. Rev. Lett. 103, 135702 (2009). Y. S. Cho et al., Phys. Rev. Lett. 103, 135702 (2009). F. Radicchi et al., Phys. Rev. Lett. 103, 168701(2009). F. Radicchi et al., Phys. Rev. Lett. 103, 168701(2009). R. M. Ziff, Phys. Rev. E 82, 051105 (2010). R. M. Ziff, Phys. Rev. E 82, 051105 (2010). L. Tian et al., arXiv:1010.5900 (2010). L. Tian et al., arXiv:1010.5900 (2010). P. Grassberger et al., arXiv:1103.3728v2. P. Grassberger et al., arXiv:1103.3728v2. F. Radicchi et al., Phys. Rev. E 81, 036110 (2010). F. Radicchi et al., Phys. Rev. E 81, 036110 (2010). S. Fortunato et al., arXiv:1101.3567v1 (2011). S. Fortunato et al., arXiv:1101.3567v1 (2011). J. Nagler et al. Nature Physics, 7, 265 (2011). J. Nagler et al. Nature Physics, 7, 265 (2011).

12. Is the explosive percolation continuous or first-order? Suggest to be continuous: R. A. da Costa et al., Phys. Rev. Lett. 105, 255701 (2010). R. A. da Costa et al., Phys. Rev. Lett. 105, 255701 (2010). O. Riordan et al., Science 333, 322 (2011). O. Riordan et al., Science 333, 322 (2011).  = 0.0555(1) ----- accuracy is questioned  = 0.0555(1) ----- accuracy is questioned

13. The Generalized Achilioptas process (GAP) Choosing two unoccupied edges randomly Choosing two unoccupied edges randomly The edge with the minimum product of the cluster sizes is taken to be connected with a probability “ p ” The edge with the minimum product of the cluster sizes is taken to be connected with a probability “ p ” p=0.5 ER model p=0.5 ER model p=1 PR model p=1 PR model  introducing the effects gradually  introducing the effects gradually

14. The largest cluster in two- dimensional lattice network under GAP

15. Finite-size scaling form of cluster sizes near critical point The largest cluster: The second largest cluster:

16. At critical point t = 0  fixed-point for different L  straight line for ln L Both properties are used to determine critical point

17. Fixed-point of s 2 /s 1 at p=0.5

18. Straight line of ln s 1 at p=0.5

19. Fixed-point of s 2 /s 1 at p=1.0

20. Straight line of ln s 1 at p=1.0

21. Summary of critical points and critical exponents

22. Finite-size scaling function of s 2 /s 1

23. Critical exponent ratios

24. Inverse of the critical exponent of correlation length

25. Ratio s 2 /s 1 at the critical point

26. The universality class of two- dimensional lattice networks (critical exponenets, ratios……) depends on the probability parameter “p”

27. Conclusion  Phase transitions in two-dimensional lattice network under GAP are continuous  Universality class of complex network depends on more than “d” and “n” depends on more than “d” and “n”  Further investigations are needed for understanding the universality class of complex systems understanding the universality class of complex systems

28. Collabotators Mao-xin Liu (ITP) Mao-xin Liu (ITP) Jingfang Fan (ITP) Jingfang Fan (ITP) Dr. Liangsheng Li Dr. Liangsheng Li (Beijign Institute of Technology) (Beijign Institute of Technology)

29. Thank you! Thank you!