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International Workshop on Complex Networks, Seoul (23-24 June 2005)
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)
I. Static model of scale-free networks
static model
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
We consider sparse, undirected, non-degenerate graphs only.
1. Degree of a vertex :
2. Degree distribution:
,1
N
i jj
aik=
=å
( ) l-:DP k k
G= adjacency matrix element
,i ja
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:
( ) ( )=åG
O P G O 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
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
Construction of the static model
= Zipf exponent
When =0 ER case. Walker algorithm (+Robin Hood method) constructs
networks in time . network in 1 min on a PC.C
om
men
ts
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
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 fProb( ) ( ) e
2=1 1 -= = - i jK NP Pij ija fProb( ) e
Each link is attached independently but with inhomegeous probability f i,j .
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
1
2 2 2 2
1 1
( ) Poissonian with mean 2 ,
/ 2 ,
/ ,
1( ) Scale-free with 1
i i i
N
ii
N N
i ii i
D
P k k KNP
k k N K
k k N k k NP
P k
Degree distribution
Percolation Transition
2
2
( 1)( 3)for 31
2( 2)2
0 for 3C
ii
KNP
International Workshop on Complex Networks, Seoul (23-24 June 2005)
static model
Recall
When (),
When ()
Com
men
ts
ln
ln
iN
ln
ln
jN
Strictly uncorrelated in links, but vertex correlation enters (for finite N) when 2<<3 due to the “fermionic constraint” (no self-loops and no multiple edges) .
21 -= - i jK NP Pijf e
, 2 /i j i j i jf KNPP k k k N
1/( 1)max 1 ~k k N
International Workshop on Complex Networks, Seoul (23-24 June 2005)
II. Vertex correlation functions
Vertex correlations
nn ( ) average degree of nearest neighbors of vertices with
degree (ANND).
( ) clustering coefficient of vertices with degree
k k
k
C k k
Related work: Catanzaro and Pastor-Satoras, EPJ (2005)
International Workshop on Complex Networks, Seoul (23-24 June 2005)
2.5
2.5
3 4
3 4
Vertex correlations
Simulation results of k nn (k) and C(k) on Static Model
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Vertex correlations
Our method of analytical evaluations:
, ,, , , ,,, ,
nn
( 1)( 1) ( 1)
( )i j j mi j j m i j j m
j i m i jj i m i j j i m i j
i i i
a aa a f f
k ik k k
11 1 ( ) ( ) ( ) ( ) (1)
N NN
mF x dx F N F m F x dx F
For a monotone decreasing function F(x) ,
1/ 2( , ~ , ~ / )j iN N x k NP k N
( 1),
,
( 1) (1 )
(1 )
xyj m
m i jx
f e y dy
O e
Use this to approximate the first sum as
Similarly for the second sum.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
3 2
1 1
23 (3 ) 1
for ~( ) ~
for ~
cnn
c
N k k Nk k
N k k k N
Vertex correlations
Result (1) for( ) 2< <3nnk k
International Workshop on Complex Networks, Seoul (23-24 June 2005)
2
2
( 2 8 7) /( 1) (3 ) 1/ 2
5 2 2(3 ) 1/ 2 1/( 1)max
ln for
( ) ~ for
for ~
C
C
N N k k
C k N k k k N
N k N k k N
Vertex correlations
Result (2) for
( ) 2< <3C k
International Workshop on Complex Networks, Seoul (23-24 June 2005)
iC010
-210
-410
-610-810
310
710
510
Result (3) Finite size effect of the clustering coefficient for 2<<3:
Vertex correlations
2
ln ( ) ~
NC C i
N
1c.f. for 3C
N
International Workshop on Complex Networks, Seoul (23-24 June 2005)
III. Number of self-avoiding walks and circuits
Number of SAWs 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
' '
, , , , , ,L i j j k l m i j j k l ma a a f f f N
where the sum is over distinct ordered set of (L+1) vertices,{ , , , }.i j m We consider finite L only.
The number of circuits or self-avoiding loops of size L on a graph is
' '
, , , , , ,L i j j k l i i j j k l ia a a f f f M
with the first and the last nodes coinciding.
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
1
, , , ,11( , , ) 1
( 1) (1 ) (1 )xy x
N N
i j i j i t ij i m j
e y dy e
f f f dt f
1 2 3 4
0.2
0.4
0.6
0.8
1
Strategy for 2< <3: For upper bounds, we use
repeatedly. Similarly for lower bounds with
The leading powers of N in both bounds are the same.
Note: The “surface terms” are of the same order as the “bulk terms”.
1
0(1(1 ) ) ( ) x
ee L x
1/ 2( , ~ , )i
N N x k N P
0
if 0 11 ( )
1 if 1x x x
e L xx
and
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
Result(3): Number of L-step self-avoiding walks
3 For , straightforward in the static model
For , the leading order terms in N are obtained.
2 3
2
( 1)2
( 1)
(3 )31 1
2( 1)2
31 1 ( 1) / 22
1 ( 3)
~ 0.25 ln ( 3)
~ (2 3, even)
~ ln (2 3, odd)
L
L
L
L
L
L
L L
L
kN k
k
N k k N
N N L
N N L
N
N
N
N
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
Typical configurations of SAWs (2<λ<3)
L=2
L=3
L=4
1/( 1) 0H H
22 H H
~ , ~
~
k N n N
k n
N
1/ 2 (3 ) / 2S S
2 2 2 23 S S H H
~ , ~
~
k N n N
k n k n
N
( 2) /( 1) (3 )SS SS
2 24 H SS SS
~ , ~
~
k N n N
k k n
N
H
H H
S S
B B
BB
B BSS
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Number of SAWs and Circuits
( )
2
(3 ) / 2
(3 ) / 2
11 ( 3)
2
~ 0.25 ln ( 3)
~ ln (2 3, even)
~ (2 3, odd)
L
L
L
L
LL
LL
k
L k
k N
N N L
N L
M
M
M
M
-
-
æ ö< > ÷ç ÷= - >ç ÷ç ÷ç< >è ø
< > =
< < =
< < =
Results(4): Number of circuits of size
( 3)L L
International Workshop on Complex Networks, Seoul (23-24 June 2005)
IV.Critical phenomena of spin models defined on the static model
Spin models on SM
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.
, , ,[ ] with quenched bond disorder i j i j i j i ji j
H G a J S S a
QQQQQQQQQQQQQQQQQQQQQQQQQQQQ
0
, , , eff
Free energy in replica method= log Z lim 1 /
tr (1 ) exp( ) tr exp[ ]
n
n
ni j i j i j i j
i j
Z n
Z f f J S S H
QQQQQQQQQQQQQQQQQQQQQQQQQQQQ
eff , ,
,
ln 1 exp( ) 1
exp( ) 1 ( )
i j i ji j
i j i j i ji j
H f J S S
k NPP J S S o N
QQQQQQQQQQQQQQQQQQQQQQQQQQQQ
QQQQQQQQQQQQQQQQQQQQQQQQQQQQ
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Spin models on SM
3
2
( ln ) for 2 3
( ) ~ ((ln ) ) for 3
(1) for 3
O N N
o N O N
O
The effective Hamiltonian reduces to a mean-field type one with infinite number of order parameters,
1 1 1
, , ,N N N
i i i i i i i i ii i i
P S P S S P S S S
QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
When J i,j are also quenched random variables, do additional averages on each J i,j .
International Workshop on Complex Networks, Seoul (23-24 June 2005)
Spin models on SM
We applied this formalism to the Ising spin-glass [DH Kim et al PRE (2005)]
, , ,( ) ( ) (1 ) ( )i j i j i jP J r J J r 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:
1
1/( 3)
1 2 2
2( 2) /(3 )
( ) ( 4)
( ) / ln( ) ( 4)
~ ( ) (3 4)
exp( 2 / ) ( 3)
(2 3)
C
C CN
i i i Ci
T T
T T T T
q P S S T T
T T k
T
2/(3 )21
1~ ~ for 2 3
N
i ii
qQ S S T
N k T
( 2) /(3 ) 1/(3 )
1 1
1~ , ~
N N
i i ii i
m P S T M S TN
To be compared with the ferromagnetic behavior for 2<<3;
International Workshop on Complex Networks, Seoul (23-24 June 2005)
V. Conclusion1. 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.