Vertex Correlations, Self-Avoiding Walks and
Critical Phenomena on the Static Model of
Scale-Free Networks
DOOCHUL KIM (Seoul National University)
Collaborators:
Byungnam Kahng (SNU), Kwang-Il Goh (SNU/Notre
Dame), Deok-Sun Lee (Saarlandes), Jae- Sung Lee
(SNU), G. J. Rodgers (Brunel) D.H. Kim (SNU)
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Outline
I.
Static model of scale-free networks
II. Vertex correlation functions
III. Number of self-avoiding walks and circuits
IV. Critical phenomena of spin models defined
on the static model
V. Conclusion
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
I. Static model of
scale-free networks
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
G
ai , j
= adjacency matrix element
(0,1)
N
ki =
1. Degree of a vertex i:
е
ai , j
j= 1
2. Degree distribution:
PD (k ) : k - l
 We consider sparse, undirected, non-degenerate graphs only.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
 For theoretical treatment, one needs to take
averages over an ensemble of graphs G. 
Grandcanonical ensemble of graphs:
O =
е
P (G)O(G)
G
 Static model: Goh et al PRL (2001), Lee et al NPB (2004),
Pramana (2005, Statpys 22 proceedings), DH Kim et al
PRE(2005 to appear)
Precursor of the “hidden variable” model [Caldarelli et al PRL
(2002), Soederberg PRE (2002) , Boguna and Pastor-Satorras
PRE (2003)]
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
 Construction of the static model
1. Each site is given a weight (“fitness”)
N
Pi = i
- m
е
j - m (i = 1,L ,N ),
j =1
е
i
Pi = 1, (0 < m< 1)
m = Zipf exponent
2. In each unit time, select one vertex i
with prob. Pi and another vertex j with prob. Pj.
3. If i=j or aij=1 already, do nothing (fermionic constraint).
Otherwise add a link, i.e., set aij=1.
Comments
4. Repeat steps 2,3 NK times (K = time = fugacity = L/N).
 When m=0  ER case.
 Walker algorithm (+Robin Hood method) constructs networks in
time O(N).
 N=107 network in 1 min on a PC.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
 Prob( ai j =0)= (1- 2Pi Pj )NK = e
- 2 K NPi Pj
= 1- f i j
 Prob( ai j =1)= f i j = 1- e
- 2 K NPi Pj
 Such algorithm realizes a “grandcanonical ensemble”
of graphs G={aij} with weights
P (G) =
Х f i j Х (1-f i j ) =
bОG
bПG
ij
f
Х i j (1- f i j )
a
1- ai j
i> j
Each link is attached independently but with inhomegeous
probability f i,j .
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
Degree distribution
P (ki ) = Poissonian with mean  ki = 2 KNPi ,
N
 k    ki  / N = 2 K ,
i =1
 N
2
 k    k  / N = k    k    NPi  ,
i =1
 i =1

1
PD (k ) = Scale-free with  = 1 
N
2
2
i
2
m
Percolation Transition
KC =
2
(
1
2
NP
i i
)
 (  1)(  3)

=  2(  2) 2
0

International Workshop on Complex Networks, Seoul (23-24 June 2005)
for   3
for   3
static model
 Recall
- 2 K NPi Pj
ln j
ln N
f i j = 1- e
Comments
 When 3 (0m1/2),
fi , j  2 KNPP
i j = ki k j /  k  N
1
3
 When 23 (1/2m1)
fij

0
fij2KNPi
Pj
fij
1
3
1
ln i
ln N
 Strictly uncorrelated in links, but vertex correlation enters (for
finite N) when 2<<3 due to the “fermionic constraint” (no selfloops and no multiple edges) .
kmax  k1 ~ N 1/(  1)
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
II. Vertex correlation functions
knn (k ) = average degree of nearest neighbors of vertices with
degree k (ANND).
C (k ) = clustering coefficient of vertices with degree k
Related work: Catanzaro and Pastor-Satoras, EPJ (2005)
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
Simulation results of k nn (k) and C(k) on Static Model
 = 2.5
 =3
=4
 = 2.5
 =3
=4
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
Our method of analytical evaluations:
knn (i ) 
a (  a
i, j
j i
mi , j
ki
j ,m
 1)

a (  a
i, j
j i
mi , j
j ,m
 1)
=
 ki 
 f (
j i
i, j
mi , j
f j ,m  1)
 ki 
For a monotone decreasing function F(x) ,

N
1
F ( x) dx  F ( N )  m=1 F (m)   F ( x) dx  F (1)
N
N
1
Use this to approximate the first sum as

m i , j
f j , m = (  1) 
(  1)


(1  e  xy ) y   dy
O (1  e  x )
(  = N m  ,  ~ N 1/ 2 , x =  k  N Pj ~  ki  / N )
Similarly for the second sum.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
knn (k ) for 2< <3
 2
 31
 1
N
for
k

k
~
N

c
knn (k ) ~ 
 2
 N 3 k  (3 )
 1
for
k

k
~
N
c

Result (1)
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
Result (2)
C (k ) for 2< <3
 N 2 ln N
for k  kC

 ( 2 2 8 7) /(  1)  (3 )
1/ 2
C (k ) ~  N
k
for kC  k  N
 5 2 2(3 )
1/ 2
1/(  1)
N
k
for
N

k

k
~
N
max


International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
Result (3) Finite size effect of the clustering
coefficient for 2<l<3:
ln N
C = C (i ) ~   2
N
Ci
100
10-2
10-4
10-6
10-8

c.f. C
1
N
for   3
International Workshop on Complex Networks, Seoul (23-24 June 2005)
2
3
10
5
3 4
10
5 6
78
107
91
0
N
Number of SAWs and circuits
III. Number of self-avoiding
walks and circuits
 The number of self-avoiding walks and circuits
(self-avoiding loops) are of basic interest in
graph theory.
 Some related works are: Bianconi and Capocci,
PRL (2003), Herrero, cond-mat (2004), Bianconi
and Marsili, cond-mat (2005) etc.
 Issue: How does the vertex correlation work on
the statistics for 2<<3 ?
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
The number of L-step self-avoiding walks on a graph is
N L   ai , j a j ,k
'
al ,m =  fi , j f j ,k
'
fl ,m
where the sum is over distinct ordered set of (L+1) vertices,
{i, j ,
We consider finite L only.
, m}.
The number of circuits or self-avoiding loops of size L on a graph is
M
  ai , j a j ,k
'
L
al ,i =  fi , j f j ,k
with the first and the last nodes coinciding.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
'
fl ,i
Number of SAWs and Circuits
Strategy for 2< <3: For upper bounds, we use

j ( i ,
N
f
fi , j 
j =1
,m )
= (  1) 
m
( = N  ,
and
 1
~N



1/ 2
,
i, j
(1  e
x=

 xy

)y
N
1

f i ,t dt  f i ,1
dy  (1  e
 k  N Pi )
 x
)
1
0.8
1 e
x
 x if 0  x  1
 L0 ( x)  
1 if 1  x
0.6
0.4
0.2
1
x
2
3
1
4
repeatedly. Similarly for lower bounds with (1  e )  (1  e ) L0 ( x )
The leading powers of N in both bounds are the same.
Note: The “surface terms” are of the same order as the “bulk
terms”.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
 For
 3
, straightforward in the static model
 For 2    3 , the leading order terms in N are obtained.
Result(3): Number of L-step self-avoiding walks
k  
N L= N  k  
 1
 k 

2
( L 1)
N L ~ N  k  ( 0.25  k  ln N )
N L~ N
N L~ N
 3  
1 
( L 1)
 2 
 3  
1 
( L 1)
 2 
N
(  3)
( L 1)
(3  ) 2
2(  1)
( ln N )
( = 3)
(2    3, L = even)
( L 1) / 2
International Workshop on Complex Networks, Seoul (23-24 June 2005)
(2    3, L = odd)
Number of SAWs and Circuits
Typical configurations of SAWs (2<λ<3)
H
kH ~ N 1/(  1) , nH ~ N 0
L=2
B
N 2 ~ kH 2 nH
B
S
S
kS ~ N 1/ 2 , nS ~ N (3 ) / 2
L=3
N 3 ~ kS nS  kH nH
2
B
B
H
H
SS
2
2
kSS ~ N (  2) /(  1) , nSS ~ N (3 )
L=4
B
2
B
N 4 ~ kH 2 kSS2 nSS
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
Results(4): Number of circuits of size
L ( L  3)
L
2
ц
1 ж
<
k
>
зз
ч
M L=
- 1ч
ч
з
ч
2L и < k >
ш
L
M L ~ (0.25 < k > ln N )
M L ~ N (3-  ) L / 2 ln N
M L~ N
(3-  ) L / 2
International Workshop on Complex Networks, Seoul (23-24 June 2005)
( > 3)
( = 3)
(2 <  < 3, L = even)
(2 <  < 3, L = odd)
Spin models on SM
IV. Critical phenomena of spin
models defined on the
static model
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Spin models on SM
Spin models defined on the static model network can be analyzed
by the replica method.
H [G] =  ai , j J i , j Si  S j
i j
with quenched bond disorder ai , j 
Free energy in replica method= log Z = lim Z n  1 / n
n 0
Z
n


 
= tr (1  fi , j )  fi , j exp(  J i , j  S i  S j )   tr exp[ H eff ]

i j 

 H eff





=  ln 1  f i , j  exp(  J i , j  S  S )  1 
i j








=   k  NPP
i j  exp(  J i , j  S i  S j )  1   o( N )
i j



International Workshop on Complex Networks, Seoul (23-24 June 2005)
Spin models on SM
O( N 3 ln N ) for 2    3

o( N ) ~ O((ln N ) 2 )
for  = 3
O(1)
for   3

When J i,j are also quenched random variables, do additional
averages on each J i,j .
The effective Hamiltonian reduces to a mean-field
type one with infinite number of order parameters,
N
 Pi  Si  ,
i =1

N
 Pi  Si Si  ,


i =1
International Workshop on Complex Networks, Seoul (23-24 June 2005)
N
  
P

S
 i i Si Si  ,
i =1
Spin models on SM
We applied this formalism to the Ising spin-glass [DH Kim et al PRE (2005)]
P( J i , j ) = r ( J i , j  J )  (1  r )  ( J i , j  J )
Phase diagrams in T-r plane for   3 and  <3
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Spin models on SM
Critical behavior of the spin-glass order parameter
in the replica symmetric solution:
(TC  T )

1
(
T

T
)
/
ln(
T

T
)
C
 C

~ (TC  T )1/(  3)
 2
2
T
exp(

2
T
/  k )

T 2(   2) /(3 )

N
q   Pi Si Si
i =1
1
Q
N

N
i =1


Si Si
(  4)
( = 4)
(3    4)
( = 3)
(2    3)
q
2 /(3  )
~
~
T
 k T2
for 2    3
To be compared with the ferromagnetic behavior for 2<<3;
N
1
m   Pi  Si ~ T  (   2) /(3 ) , M 
N
i =1
International Workshop on Complex Networks, Seoul (23-24 June 2005)
N
1/(3  )

S

~
T
 i
i =1
V. Conclusion
1. The static model of scale-free network allows detailed analytical
calculation of various graph properties and free-energy of
statistical models defined on such network.
2. The constraint that there is no self-loops and multiple links
introduces local vertex correlations when , the degree exponent,
is less than 3.
3. Two node and three node correlation functions, and the number
of self-avoiding walks and circuits are obtained for 2< <3. The
walk statistics depend on the even-odd parity.
4. The replica method is used to obtain the critical behavior of the
spin-glass order parameters in the replica symmetry solution.
International Workshop on Complex Networks, Seoul (23-24 June 2005)