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)


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


I. Introduction

introduction


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


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


II. Static model of scale-free networks

static model


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)


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.


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


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


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


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


- 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


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.



- 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



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



- 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



- 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


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



,( )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



V. Spectral density of adjacency and related matrices

Spectral density


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


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.


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


Spectral density

( ) versus for 2.5, 3.0, 4.0, and (ER)A E E


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…


( ) versus for 2.5, 3.0, 4.0, and 10L


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


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


VI. Ising spin-glass transitions

Spin models on SM


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 .


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.


Spin models on SM

Phase diagrams in T-r plane for 3 and<3


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;


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

application of replica method to scale-free networks: spectral density and spin-glass transition

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