kps 2007 (april 19, 2007) on spectral density of scale-free networks doochul kim (department of...
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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
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I. Introduction
introduction
Many real world networks are Many real world networks are scale-free…scale-free…
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Internet is a network
Nodes: Routers for IR network,
Autonomous Systems for AS network
Links: physical lines
introduction
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introductionInternet is scale-free
Evolution of degree distribution Evolution of degree distribution of AS of AS
2.1
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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
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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
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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.
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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.
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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)]
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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
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- 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
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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
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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.