Phase Transitions of Complex Phase Transitions of Complex Networks and relatedNetworks and related

Xiaosong ChenXiaosong Chen

Institute of Theoretical PhysicsInstitute of Theoretical PhysicsChinese Academy of SciencesChinese Academy of Sciences

CPOD-2011, WuhanCPOD-2011, Wuhan

OutlineOutline Phase transitions and critical phenomena,Phase transitions and critical phenomena, universality and scalinguniversality and scaling

Complex networks and percolation phase Complex networks and percolation phase transitiontransition

Phase transitions of two-dimensional lattice Phase transitions of two-dimensional lattice networks under a generalized networks under a generalized

AP process AP process

ConclusionsConclusions

Phase diagram of normal fluidsPhase diagram of normal fluids

Phase transitions and critical Phase transitions and critical phenomenaphenomena

Phase:Phase: homogenous, equilibrium, macroscopic scale.homogenous, equilibrium, macroscopic scale. For example: For example: gases, liquids, solids, plasma,……gases, liquids, solids, plasma,……

Phase transitions:Phase transitions:discontinuous:discontinuous: abrupt changeabrupt change of order parameter of order parameter (first-order)(first-order)continuous:continuous: continuous changecontinuous change of order parameter of order parameter (critical )(critical ) divergent divergent response functions ← response functions ← correlation lengthcorrelation length Gases Gases → → plasma : no phase transitionplasma : no phase transition

Scaling and universality in Scaling and universality in critical phenomenacritical phenomena

Correlation length: ∞0t|

t = (T-Tc)/Tc

ScalingScaling ：： ffss(t,h) = A(t,h) = A11 t t ddW ( AW ( A22h t h t ))

Finite-size scalingFinite-size scaling ： ： ffss(t,h,L) = L (t,h,L) = L --ddY (t L Y (t L h L h L ))

UniversalityUniversality

critical exponents, scaling functions … critical exponents, scaling functions …

depend only ondepend only on (d,n) (d,n) d: d: dimensionality of systemdimensionality of system n: n: number of order parameter componentsnumber of order parameter components

Irrelevant Irrelevant with microscopic details of systemswith microscopic details of systems

Complex networksComplex networks

consist of:consist of: nodesnodes edges edges

examples: randomexamples: random lattice lattice scale free scale free small world……small world……

Percolation phase transition in Percolation phase transition in networksnetworks

Begin with “N” isolated nodesBegin with “N” isolated nodes

“ “m” edges are added (different ways)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 size of the largest cluster / N finite finiteemergence of a new phase emergence of a new phase percolation transition percolation transition

For a review: Rev. Mod. Phys. 80, 1275 (2008)For a review: Rev. Mod. Phys. 80, 1275 (2008)

Percolation phase transition of Percolation phase transition of random networkrandom network

(emergence of a giant cluster )(emergence of a giant cluster )

First-order phase transition

The Achilioptas processThe Achilioptas process

Choosing two unoccupied edges Choosing two unoccupied edges randomlyrandomly

The edge with the minimum product The edge with the minimum product of the cluster sizes is connected.of the cluster sizes is connected.

Is the explosive percolation Is the explosive percolation continuous or first-order? continuous or first-order?

Support to be first-order transition: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).

Is the explosive percolation Is the explosive percolation continuous or first-order? continuous or first-order?

Suggest to be continuous:Suggest to be continuous: R. A. da Costa et al., Phys. Rev. Lett. 105, 2557R. A. da Costa et al., Phys. Rev. Lett. 105, 2557

01 (2010).01 (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

The Generalized Achilioptas proceThe Generalized Achilioptas process (GAP)ss (GAP)

Choosing two unoccupied edges Choosing two unoccupied edges randomlyrandomly

The edge with the minimum product The edge with the minimum product of the cluster sizes is taken to be of the cluster sizes is taken to be connected with a probability “p”connected with a probability “p”

p=0.5 ER modelp=0.5 ER model p=1 PR modelp=1 PR model introducing the effects graduallyintroducing the effects gradually

The largest cluster in two-The largest cluster in two-dimensional lattice network under dimensional lattice network under

GAPGAP

Finite-size scaling form of cluster Finite-size scaling form of cluster sizes near critical pointsizes near critical point

The largest cluster:

The second largest cluster:

At critical point t = 0At critical point t = 0

fixed-point for different L

straight line for ln L

Both properties are used

to determine critical point

Fixed-point of sFixed-point of s22 /s /s11 at p=0.5 at p=0.5

Straight line of ln sStraight line of ln s11 at p=0.5 at p=0.5

Fixed-point of sFixed-point of s22 /s /s11 at p=1.0 at p=1.0

Straight line of ln sStraight line of ln s11 at p=1.0 at p=1.0

Summary of critical points and Summary of critical points and critical exponentscritical exponents

Finite-size scaling function of sFinite-size scaling function of s22 /s /s11

Critical exponent ratiosCritical exponent ratios

Inverse of the critical exponent of Inverse of the critical exponent of correlation lengthcorrelation length

Ratio sRatio s22/s/s11 at the critical point at the critical point

The universality class of two-dimensioThe universality class of two-dimensional lattice networksnal lattice networks

(critical exponenets, ratios……)(critical exponenets, ratios……)

depends on depends on

the probability parameter “p”the probability parameter “p”

ConclusionConclusion

Phase transitions in two-dimensional lattice nePhase transitions in two-dimensional lattice network under GAP are continuoustwork under GAP are continuous Universality class of complex networkUniversality class of complex network depends on more than “d” and “n”depends on more than “d” and “n” Further investigations are needed forFurther investigations are needed for understanding the universality class of compleunderstanding the universality class of complex systemsx systems

CollabotatorsCollabotators Mao-xin LiuMao-xin Liu (ITP)(ITP)

Jingfang FanJingfang Fan (ITP)(ITP)

Dr. Liangsheng LiDr. Liangsheng Li (Beijign Institute of Technology) (Beijign Institute of Technology)

Thank you!Thank you!