KPS 2007 (April 19, 2007)

On spectral density of scale-free networks

Doochul Kim (Department of Physics and Astronomy, Seoul National University)

Collaborators:

Byungnam Kahng (SNU), Geoff. J. Rodgers (Brunel, UK)

KPS 2007 (April 19, 2007)

Outline

I. Introduction

II. Matrices of Interest

III. An Equilibrium Ensemble of Scale-Free Graphs – The Static Model

IV. Replica Method: General formalism

V. Spectral Densities in the Dense Graph Limit

VI. Summary and Discussion

KPS 2007 (April 19, 2007)

I. Introduction

introduction

Many real world networks are Many real world networks are scale-free…scale-free…

KPS 2007 (April 19, 2007)

Internet is a network

Nodes: Routers for IR network,

Autonomous Systems for AS network

Links: physical lines

introduction

KPS 2007 (April 19, 2007)

introductionInternet is scale-free

Evolution of degree distribution Evolution of degree distribution of AS of AS

2.1

KPS 2007 (April 19, 2007)

introductionInternet is scale-free

Load (or Betweenness Load (or Betweenness Centrality) distribution oCentrality) distribution of AS, AS+ and IR networf AS, AS+ and IR networksks

( )

2.0

P b b

KPS 2007 (April 19, 2007)

http://www.nd.edu/~networks/gallery.htm

There are many more examples that are scale-free approximately…..

Information networks (WWW)

Biological networks (Protein interaction network)

Social network (Collaboration Network)

introduction

KPS 2007 (April 19, 2007)

introduction

• We consider sparse, undirected, simple graphs (no self-loops, no multiple bonds) with N nodes and L links (2L/N=p).

• Degree of a vertex i:

• SF degree distribution:

G= adjacency matrix with element 1 if connected and 0 otherwise

,i jA

( )dP d d

,1

N

i i jj

d A

introduction

KPS 2007 (April 19, 2007)

introduction

• Spectral properties of matrices defined on such scale-free networks are of interest.

• For theoretical treatment, one needs to take averages of dynamic quantities over an ensemble of graphs.

( ) ( )G

O O G P G

• One can apply the replica method to obtain the spectral density of a class of scale-free networks, in the dense graph limit after the thermodynamic limit.

KPS 2007 (April 19, 2007)

II. Matrices of Interest

matrices of interest

We consider 5 types of We consider 5 types of matrices associated with a matrices associated with a graph G.graph G.

KPS 2007 (April 19, 2007)

matrices of interest

,

, , ,

,,

1. (1 or 0)

2.

( )/

( mean degree = 2 link d

adjacency matrix

Laplacian

random walk matri

ensity)

3.

similar to

x

i j

i j i i j i j

i ji j

i j

A

L d A p

p

AR

d d

,

(prob. of a walk from node to node .

If a node is isolated, no move.)

i j

i

A

d

i j

KPS 2007 (April 19, 2007)

matrices of interest

, ,,

, ,,

4.

similar to

( = vertex dependent constant)

5.

weighted adjacency matrix

weighted Lapla

simi

cian

i j i ji j

ii j

i i

i i j i ji j

i j

A AB

qq q

q d

d AW

q q

, ,lar to

( = vertex dependent constant)

i i j i j

i

i i

d A

q

q d

KPS 2007 (April 19, 2007)

III. An Equilibrium Ensemble of Scale-Free Graphs – The Static model

static model

The static model is an efficient The static model is an efficient way of generating the scale-way of generating the scale-free network with arbitrary free network with arbitrary expected degree sequences. expected degree sequences.

KPS 2007 (April 19, 2007)

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 Chung-Lu model (2002)

- Closely related to the “hidden variable” models [Caldarelli et al PRL (2002), Boguna and Pastor-Satorras PRE (2003), Park-Newman (2003)]

KPS 2007 (April 19, 2007)

static model

1. Each site is given a weight (“fitness”)

2. 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 (<L>=Np/2, p= fugacity for links).

Construction of the static model

1/( 1) ( 1,..., ), 1 , ( 2)i iiP i i N P

KPS 2007 (April 19, 2007)

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

KPS 2007 (April 19, 2007)

static model

1

( ) Poissonian

expected degree sequence

/ mean degree

1( ) as

i

i i

N

ii

d

P d

d pNP

d d N p

P d dd

- Degree distribution

- Percolation Transition

2

2

( 1)( 3)for 31

( 2)

0 for 3C

ii

pNP

KPS 2007 (April 19, 2007)

- c.f. Chung-Lu model:

, < 1i j i jf w wwith (expected degree sequence),

and 1/ 1/

i i

ii

w d

w pN

1/ 2

max for 2 3d N

1/( 1)maxc.f. ~ in the static modeld N

static model

- Erdos-Renyi model :

KPS 2007 (April 19, 2007)

IV. Replica method: General formalism

Replica method: General formalism

The replica method may be The replica method may be applied to perform the graph applied to perform the graph ensemble averages in the ensemble averages in the thermodynamic limit. thermodynamic limit.

KPS 2007 (April 19, 2007)

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

KPS 2007 (April 19, 2007)

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

::::::::::::::::::::::::::::

KPS 2007 (April 19, 2007)

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 rigorously that the remainder R/N is small in the thermodynamic limit for the equilibrium ensembles mentioned. E.g., for the static model, (PRE 2005)

3

2

( ln ) for 2 3

((ln ) ) for 3

(1) for 3

O N N

R O N

O

KPS 2007 (April 19, 2007)

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 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::

- 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

::::::::::::::

- 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

KPS 2007 (April 19, 2007)

1

tr exp ( ) ( )n

i ih pNP g

::::::::::::::

,

tr ( ) exp ( ) ( )

( )

tr exp ( ) ( )

R i

Ri

i

O h pNP g

O

h pNP g

::::::::::::: :

::::::::::::::

1 ,

( ) exp ( , ) 1n

ii i

g P V

- The single site partition function is

- The effective “mean-field energy” function inside is determined via the non-linear functional equation:

Replica method: General formalism

KPS 2007 (April 19, 2007)

,( )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 .

Replica method: General formalism

KPS 2007 (April 19, 2007)

V. Spectral Densities in the Dense Graph Limit

Spectral density

Formal expressions of the Formal expressions of the spectral density (the density spectral density (the density of states) are obtained for of states) are obtained for various matrices. Explicit various matrices. Explicit analytical results are analytical results are obtained in the large p limit.obtained in the large p limit.

KPS 2007 (April 19, 2007)

with eigenvalues

( )Q d is the ensemble average of density of states

d for real symmetric N by N matrix Q .

It can be calculated from the formula

2,

1 , 1

2( ) Im ln exp

2

( Im 0 )

N N

Q k k k l lk k l

iD Q

N

Spectral density

KPS 2007 (April 19, 2007)

Spectral density

- Apply the previous formalism to the adjacency matrix ,i jA

2

0

1 210

1( ) Re exp ( )

2

with

( ) ( ) ( ) exp ( )2

A kk

k kk

iy y d g y dy

N

ig x pN d x J xy y d g y dy

- Analytic treatment is possible in the dense graph limit:p

KPS 2007 (April 19, 2007)

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 et al. 2003, Rodgers et al. JPA 2005)

1ER limit ( ): semi-circle law ( ) 1

4

2 : analytic maximum at 0 and fat tail ( )

A

A

EE

E E

KPS 2007 (April 19, 2007)

Spectral density

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

KPS 2007 (April 19, 2007)

Spectral density

, ,,

2

L 2 1

-- For the and in the limit,

( 1)( ) Im [1, ; 1; ( 1) /( 2)]

( 2)

20 for 0

1 2

constant for

Lap

1

lacian i i j i ji j

d aL p

p

F z

Similarly…Similarly…

KPS 2007 (April 19, 2007)

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

KPS 2007 (April 19, 2007)

1/ 2 1, , ,

R

-- For the

( ) =similar to ,

( ) semi-circle law for all w

random walk

ith

matr

ix

i j i j i j i i jR d d A d A

E E p

KPS 2007 (April 19, 2007)

Spectral density

, ,, / 2 / 2

1/ 2

-- For the with =< > ,

similar to ,

- 1 ;

( ; ) ( ; ) with and

weighted adjacency mat

1

x

ri i i

i j i ji j

i j i

B A

q d

A AB

d d d

E E E p

(2 1) /(1 ) ( ; ) as B E E E

KPS 2007 (April 19, 2007)

KPS 2007 (April 19, 2007)

Spectral density

1 1 1/ 2

2

22 1

1- 1; ( 2) ( 1) and ,

1

( ) Im ( ) , where ( ) is the solution of

1, 1; 2; / ( 1) /

s

B

s s s s

E p

EE b E b E

F b b E

2

( proved by Chung-Lu-Vu (PNAS 2003) for all )

1- =1 : semi-circle law in for all : ( ) 1 / 4

B

p

E p E E

KPS 2007 (April 19, 2007)

KPS 2007 (April 19, 2007)

Spectral density

, ,

, / 2 / 2

, ,

1s

W

-- For the with =< > ,

= similar to ,

1with and , and

1 1

- 1 ;

weighted Laplacian matrix

i j i j

i j

i j i

i i

i i j i i jd d

d d d

q d

A A

E p

1

( ; ) ( ; ) for 1

- 1 ; ( ; ) ( 1)

but with ( 1) , non-trivial results obtained

- 1 ; ( ; ) for 0< 1

s

W L

W

W

E E E E

E E

E p E

E E E

KPS 2007 (April 19, 2007)

KPS 2007 (April 19, 2007)

KPS 2007 (April 19, 2007)

KPS 2007 (April 19, 2007)

VI. Summary and Discussion

• The replica method is formulated for a class of scale-free graph ensembles where each link is attached independently.

• General formula for the spectral densities of adjacency, Laplacian, random walk, weighted adjacency and weighted Laplacian matrices are obtained for sparse graphs (p=2L/N finite) in the thermodynamic limit.

• The spectral densities are obtained analytically in the large p limit.

• These results are expected to be a good approximation for 1 << p << N

KPS 2007 (April 19, 2007)

• The spectral densities at finite p, and/or finite N are unsolved problems except for special cases.

• The so-called eigenratio R for the weighted Laplacian can be estimated as ln R = |1-beta| ln N /(lambda-1) .

• The Laplacian of the weighted network is a different problem that cannot be applied here. But several steps of approximations lead it to the weighted Laplacian treated in this work.