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 -
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