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Phase Transitions of Complex Networks and related

Phase Transitions of Complex Networks and related. Xiaosong Chen Institute of Theoretical Physics Chinese Academy of Sciences CPOD-2011, Wuhan. Outline.  Phase transitions and critical phenomena, universality and scaling  Complex networks and percolation phase transition

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Phase Transitions of Complex Networks and related

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  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  Complex networks and percolation phase transition  Phase transitions of two-dimensional lattice networks under a generalized AP process  Conclusions

  3. Phase diagram of normal fluids

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

  5. Scaling and universality in critical phenomena Correlation length: x = x0 |t|-n ∞ t = (T-Tc)/Tc  Scaling： fs(t,h) = A1 t dW ( A2h t -bd )  Finite-size scaling： fs(t,h,L) = L -dY (t L 1/, h L / )

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

  7. Complex networks consist of:  nodes  edges examples: random lattice scale free 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” large enough: 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 • 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). • Y. S. Cho et al., Phys. Rev. Lett. 103, 135702 (2009). • F. Radicchi et al., Phys. Rev. Lett. 103, 168701(2009). • R. M. Ziff, Phys. Rev. E 82, 051105 (2010). • L. Tian et al., arXiv:1010.5900 (2010). • P. Grassberger et al., arXiv:1103.3728v2. • F. Radicchi et al., Phys. Rev. E 81, 036110 (2010). • S. Fortunato et al., arXiv:1101.3567v1 (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). • O. Riordan et al., Science 333, 322 (2011).  = 0.0555(1) ----- accuracy is questioned

  13. The Generalized Achilioptas process (GAP) • Choosing two unoccupied edges randomly • 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=1 PR model  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 s2 /s1 at p=0.5

  18. Straight line of ln s1 at p=0.5

  19. Fixed-point of s2 /s1 at p=1.0

  20. Straight line of ln s1 at p=1.0

  21. Summary of critical points and critical exponents

  22. Finite-size scaling function of s2 /s1

  23. Critical exponent ratios

  24. Inverse of the critical exponent of correlation length

  25. Ratio s2/s1 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”  Further investigations are needed for understanding the universality class of complex systems

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

  29. Thank you!

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