Download - Application of replica method to scale-free networks: Spectral density and spin-glass transition
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Application of replica method to scale-free networks:
Spectral density and spin-glass transition
DOOCHUL KIM (Seoul National University)
Collaborators:
Byungnam Kahng (SNU), G. J. Rodgers (Brunel), D.-H. Kim (SNU), K. Austin (Brunel), K.-I. Goh (Notre Dame)
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Outline
I. Introduction
II. Static model of scale-free networks
III. Other ensembles
IV. Replica method – General formalism
V. Spectral density of adjacency and related matrices
VI. Ising spin-glass transition
VII. Conclusion
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
I. Introduction
introduction
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
introduction
• We consider sparse, undirected, non-degenerate graphs only.
• Degree of a vertex i:
• Degree distribution:
G= adjacency matrix element (0,1)
,i ja
( )dP k k
,1
N
i i jj
k a
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
introduction
• Statistical mechanics on and of complex networks are of interest where fluctuating variables live on every vertex of the network
• For theoretical treatment, one needs to take averages of dynamic quantities over an ensemble of graphs
( ) ( )G
O O G P G
• This is of the same spirit of the disorder averages where the replica method has been applied.
• We formulate and apply the replica method to the spectral density and spin-glass transition problems on a class of scale-free networks
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
II. Static model of scale-free networks
static model
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
static model
- Static model [Goh et al PRL (2001)] is a simple realization of a grand-canonical ensemble of graphs with a fixed number of nodes including Erdos-Renyi (ER) classical random graph as a special case.
- Practically the same as the “hidden variable” model [Caldarelli et al PRL (2002), Boguna and Pastor-Satorras PRE (2003)]
- Related models are those of Chung-Lu (2002) and Park-Newman (2003)
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
static model
1. Each site is given a weight (“fitness”)
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.
4. Repeat steps 2,3 Np/2 times (p/2= time = fugacity = L/N).
Construction of the static model
1/( 1) ( 1,..., ), 1 , ( 2)i iiP i i N P
When λ is infinite ER case (classical random graph).
Walker algorithm (+Robin Hood method) constructs networks in time O(N). N=107 network in 1 min on a PC.
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
static model
Such algorithm realizes a “grandcanonical ensemble” of graphs
Each link is attached independently but with inhomegeous probability f i,j .
, ,Prob ( 1) 1 e i jpNPPi j i ja f
, ,1
, ,( ) (1 )i j i ja a
i j i ji j
P G f f
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
static model
1
( ) Poissonian
expected degree sequence
/ mean degree
( ) Scale-free with the degree exponent
i
i i
N
ii
d
P k
k pNP
k k N p
P k
- Degree distribution
- Percolation Transition
2
2
( 1)( 3)for 31
( 2)
0 for 3C
ii
pNP
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
static model
- 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) .
Recall
When λ>3,
When 2<λ<3
0 1
1
ln
ln
iN
ln
ln
jN
3-λ
3-λ
fijpNPiPj
fij1
,i j i jf pNPP
,i jf
, 1 exp[ ]i j i jf pNPP
1/( 1)max 1 ~k k N
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
III. Other ensembles
Other ensembles
- Chung-Lu model
, < 1i j i jf w wwith (expected degree sequence),
and 1/ 1/
i i
ii
w k
w pN
1/ 2
max for 2 3k N
- Static model in this notation
, 1 exp( )i j i jf w w
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
- Park-Newman Model
, with ( )1
i ji j
i j
wwf p w w
ww
- Caldarelli et al, hidden variable model
, with ( )i j i jf w w p w w
Other ensembles
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
IV. Replica method: General formalism
Replica method: General formalism
– Issue: How do we do statistical mechanics of systems defined on complex networks?
– Sparse networks are essentially trees.
– Mean field approximation is exact if applied correctly.
– But one would like to have a systematic way.
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Replica method: General formalism
- Consider a hamiltonian of the form (defined on G)
- One wants to calculate the ensemble average of ln Z(G)
- Introduce n replicas to do the graph ensemble average first
0
1ln ( ) lim
n
n
ZZ G
n
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Replica method: General formalism
The effective hamiltonian after the ensemble average is
- Since each bond is independently occupied, one can perform the graph ensemble average
eff , ,1 1
,1
( ) ln 1
with ( , ) exp ( , ) 1
N n
i i j i ji i j
n
i j i j i j
H h f S
S V
CCCCCCCCCCCCCCCCCCCCCCCCCCCC
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Replica method: General formalism
- Under the sum over {i,j}, , 1i j i jf pNPP in most cases
- So, write the second term of the effective hamiltonian as
, , ,ln(1 )i j i j i j i ji j i j
f S pNPP S R
- One can prove that the remainder R is small in the thermodynamic limit. E.g. for the static model,
3
2
( ln ) for 2 3
((ln ) ) for 3
(1) for 3
O N N
R O N
O
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Replica method: General formalism
- The nonlinear interaction term is of the form
, ( , ) ( ) ( )i j i j R R i R jR
S a O O CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
- So, the effective hamiltonian takes the form
2
eff1 1
1( ) ( )
2
N n
i R i R ii R i
H h pN a PO
CCCCCCCCCCCCCC
- Linearize each quadratic term by introducing conjugate variables QR and employ the saddle point method
2
1
1ln ln
2
Nn
R R iR i
Z pN a Q
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
1
tr exp ( ) ( )n
i ih pNP g
CCCCCCCCCCCCCC
,
tr ( ) exp ( ) ( )
( )
tr exp ( ) ( )
R i
Ri
i
O h pNP g
O
h pNP g
CCCCCCCCCCCCC C
CCCCCCCCCCCCCC
1 ,
( ) exp ( , ) 1n
ii i
g P V
- The single site partition function is
- The effective “mean-field energy” function inside is determined self-consistently
Replica method: General formalism
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
,( )R i R
ii
Q P O
- The conjugate variables takes the meaning of the order parameters
- How one can proceed from here on depends on specific problems at hand.
- We apply this formalism to the spectral density problem and the Ising spin-glass problem
Replica method: General formalism
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
V. Spectral density of adjacency and related matrices
Spectral density
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
with eigenvalues
( )M d is the ensemble average of density of states
d for real symmetric N by N matrix M
It can be calculated from the formula
2,
1 , 1
2( ) Im ln exp
2
( Im 0 )
N N
M k k k l lk k l
iD M
N
Spectral density
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spectral density
- Apply the previous formalism to the adjacency matrix ,i ja
2
0
210
1( ) Re exp ( )
2
with
( ) ( ) exp ( )2
A k kk
k kk
iP y y pNP g y dy
N
ig x P x J xy y pNP g y dy
- Analytic treatment is possible in the dense graph limit.
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spectral density
2
2 22 1
The dense graph limit
with the scaling variable / fixed:
( ) Im ( ) , where ( ) is the solution of
1, 1; ; ( 1) / ( 2)
A
p
E p
EE b E b E
F z b E b
2
(2 1)
(c.f. Chung-Lu-Vu 2003, Dorogovtsev-Goltsev-Mendes-Samukin 2003)
1ER limit ( ): semi-circle law ( ) 1
4
2 : analytic maximum at 0 and fat tail ( )
A
A
EE
E E
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spectral density
( ) versus for 2.5, 3.0, 4.0, and (ER)A E E
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spectral density
, ,,
L
-- For the Laplacian and in the limit,
20 for 0
1( )2
constant for 1
i i j i ji j
k aL p
p
Similarly…Similarly…
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
( ) versus for 2.5, 3.0, 4.0, and 10L
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
1 1/ 2, , ,
R
-- For the random walk matrix ( ) ,
( ) semi-circle law for all with
i j i i j i j i jR k a k k a
E E p
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spectral density
, ,, / 2 / 2
/ 2, ,
2C
( pro
-- For the normalized interaction matrix
,
- c.f. ( )
1- =1 : semi-circle law in for all : ( ) 1 / 4
i j i ji j
i ji
i j i j i j
a aC
k kk
C a k k
E p E E
1/ 2
(2 1) /(1 )
ved by Chung-Lu-Vu (PNAS 2003) for all )
- 0 1 ;
( ; ) ( ; ) with and 1
( ; ) as
C A
C
p
E E p
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
VI. Ising spin-glass transitions
Spin models on SM
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spin models on SM
Spin models defined on the static model SF network can be analyzed by the replica method in a similar way.
For the spin-glass model, the hamiltonian is
, , ,[ ] with quenched bond disorder i j i j i j i ji j
H G a J S S a
CCCCCCCCCCCCCCCCCCCCCCCCCCCC
, , ,and ( ) ( ) (1 ) ( )i j i j i jP J r J J r J J
J i,j are also quenched random variables, do additional averages on each J i,j .
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spin models on SM
- The effective Hamiltonian reduces to a mean-field type one with an 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
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
- Generalization of Viana and Bray (1985)’s work on ER
- Work within the replica symmetric solution.
- They are progressively of higher-order in the reduced temperature near the transition temperature.
- Perturbative analysis can be done.
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
Spin models on SM
Phase diagrams in T-r plane for 3 and<3
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
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 )EA 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<λ<3;
Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)
VII. Conclusion
• The replica method is formulated for a class of graph ensembles where each link is attached independently and is applied to statistical mechanical problems on scale-free networks.
• The spectral densities of adjacency, Laplacian, random walk, and the normalized interaction matrices are obtained analytically in the scaling limit .
• The Ising spin-glass model is solved within the replica symmetry approximation and its critical behaviors are obtained.
• The method can be applied to other problems.
p