Evolution of Scale-Free Random Graphs:

Potts Model Formulation

STATPHYS22, Bangalore July 5, 2004

DOOCHUL KIM (Seoul National University)

Collaborators: Byungnam Kahng (SNU)Kwang-Il Goh (SNU)Deok-Sun Lee (Saarlandes)

Plan

I. Introduction

II. Static model

III. Potts model representation

IV. Thermodynamic limit

V. Finite-size effect

VI. Conclusion

Introduction

• Scale-free (SF) networks: frequently encountered in Nature.

• We consider sparse, undirected, non-degenerate graphs only.

1. Degree of a vertex :[ = adjacency matrix element ]

2. Degree distribution:

( , , )= =å Li ijjk a i N1

( ) l-:DP k k

Newman et al. PRE (2001)Burda et al. PRE (2001); (2003a,b)Goh et al. PRL (2001)Caldarelli et al. PRL (2002)Chung & Lu Ann. Combinat. (2002)Söderberg PRE (2002)Berg & Lässig PRL (2002)Dorogovtsev et al. Nucl. Phys. B (2003)Park & Newman PRE (2003); cond-mat/0405566Farkas et al. cond-mat/0401640

Lee et al. Nucl. Phys. B (2004)

– Recent references

Introduction

• Ensemble of graphs:

• Various equilibrium ensembles for SF networks

– Microcanonical ensemble with given degree sequence .– Canonical ensemble with fixed .– Grandcanonical ensemble with fixed but with fluctuating .

Goh et al. PRL (2001)

Lee et al. Nucl. Phys. B (2004)

( ) ( )=åG

O P G O G

Static Model

• “Evolution of Random Graphs” by Erdős-Rényi (ER) Links are connected with equal probability. Equivalently, links are randomly attached one by one. Percolation transition when (appearance of the giant

cluster).

1. Each site is given a weight (“fitness”)

2. In each unit time, select one vertex with prob. and another vertex with prob. .

3. If or already, do nothing (fermionic constraint).Otherwise add a link, i.e., set .

4. Repeat steps 2,3 times (= time = fugacity = ).

1

( 1, , ), 1, (0 1)m m m- -

=

= = = < <å åLN

i iij

P i j i N P

visualization demo

• Static model of Goh et al. Evolution of SF random graphs

= Zipf exponent =

Static Model

– When =0 ER case.– Walker algorithm (+Robin Hood method) constructs networks in tim

e . network in 1 min in a PC.

– Monte Carlo simulation with edge addition (deletion) prob. () equivalent but inefficient.

Com

men

ts

Such algorithm realizes a “grandcanonical ensemble” of graphs ={} with weights

1( ) (1 ) -

Î Ï >

= = -Õ Õ Õ ij ija aij ij ij ij

b G b G i j

P G ff ff(1- )

• 2=0 1 2 1-= - = = -i jK NP PNKij i j ija P P f( ) ( ) eProb

• 2=1 1 -= = - i jK NP Pij ija fProb( ) e

Static Model

– Recall

–

– When (), .

– When ()

2 /( )im m:i jK NP P N j2 -1

Com

men

ts

ln

ln

iN

ln

ln

jN

– Bosonic model (allow multiple links)Prob(=) is Poissonian with =2.

21 -= - i jK NP Pijf e

Potts spin: .

Partition function:

#Z q of clusters

Potts model order parameter:

1

( ,1) 1 ( = )

1i

qi

qS mN

q

giant cluster sizeH

( )q

L N Nq

= mean number of independent loops=1

Z

Potts model representation

Potts model susceptibility mean cluster size ( ) /s

s s n s N2’1

q

• Potts model Bond percolation through Kasteleyn construction

2 2[ (1 ) ( , )]i j i jK NP P K NP Pi j

i j

e e e

- H

[(1 ) ( , )]

(1 ) [ ( , )]

( ) ( , )

b b

ij ij i ji j

ij ij i jG G G

i jG b G

ff

ff

P G

Boltzmann weight:

2 [ ( , ) 1]

H i j i ji j

K N P PHamiltonian:

Potts model representation

1) is NOT the Potts model on a scale-free network, but on the complete graph as a tool to generate SF networks.

2) It is NOT the Hamiltonian defining the equilibrium ensemble In our case,

Com

men

ts

ln ( )H P Gln( 1)

H i jK NP Pij

i j

a 2e constant

H

0

1

[ ( ,1) 1] :

( ,1) 11

1

ii

iq

i

h q

qN q

Add to

H

H

( )= ( )/P s sn s N

Generating function of

the cluster size distribution

Thermodynamic limit

Exact analytic evaluation of the Potts free energy:

• Explicit evaluations of thermodynamic quantities.

1. Vector spin representation

2. Integral representation of the partition function

3. Saddle-point analysis

1b= 1g= 3n=

sm l

1

3b

l=

-1g= 1

3

ln

l-

=-sm l

1

3b

l=

-1

3

ln

l-

=-sm l

2

~æ ö÷ç ÷ç ÷÷çè øå

uri i

i

P sH

1 2(2 ) 2-= « á ñá ñ=åc ii

K N P k k2• Percolation transition at

Thermodynamic limit

Cluster size distributions

Branching process approach [cf. Newman et al. 2001] becomes exact (almost no loops in finite clusters).

/1( )= ( )/ ~ cs sP s sn s N s e 1/

~ 1cc

Ks

K

5 14

2 22 3 3

3 42 2

32 3

2

t s

l

l ll

l ll

l ll

>

- -< <

- --

< <-

cf. Cohen et al. (2002) for site percolation

• Giant cluster size at 1/( 1)~ t -S N

Finite-size effect

1. Finite size scaling for (I) and (II) c

c

K KK-

D =

( )/ 1/m N NI,IIb n n-= Y D ( )1/ 1/s N NI,II

n n= F D ( )1/NN I,II

1 n= W Dl

4.8l =

3.6l =

Finite-size effect

2. Double peaks:

2.4l =

Double peaks in

s

3

11

2

1/

0.1

ll--:

:p

p

K N

K N

Clustering coefficient

6(# of )

(# of )ÙV

010

-210

-410

-610-810

-810

310

710

510

iC010

-210

-410

-610-810

310

710

510

3. Clustering coefficient:

Finite-size effect

Conclusion

− A simple algorithm (“static model”) is introduced to generate SF graphs.

− Corresponding equilibrium ensemble is identified and is related to the Potts model with inhomogeneous interactions on complete graph.

− Several explicit analytical results are obtained in the thermodynamic limit.

− Finite-size scaling near the percolation transition is constructed & tested numerically.

− Evolution of the SF random graphs in “time,” i.e., as the # of links increases, shows distinct behaviors for .

− Some statistical mechanical problems on the static model can be handled analytically (Work in progress).

Conclusion

− A simple algorithm (“static model”) is introduced to generate SF graphs.

− Corresponding equilibrium ensemble is identified and is related to the limit of the Potts model with inhomogeneous interactions on a complete graph.

− Explicit analytical results are obtained in the thermodynamic limit.

− Finite-size scaling near the percolation transition is constructed & tested numerically.

− Evolution of the SF random graphs in “time,” i.e., as the # of links increases, shows distinct behaviors for .

− Some statistical mechanical problems on the static model can be handled analytically (Work in progress).

lK cK uKscale-free

percolating phase

(1)O

lK pK 2 uK

scale-free

percolating phase

( )-O N (1)O

( )O N

[ , ]c pK K 1

( )N -O3

1

ll--

2

1

ll--

( )O N2

1

ll--

1

1l -

( )-O N1

1l -