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Structural correlations and critical phenomena of random scale-free networks: Analytic approaches. DOOCHUL KIM (Seoul National University). Collaborators:

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slide1

Structural correlations and critical phenomena of random scale-free networks:

Analytic approaches

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)

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide2

Outline

  • Static model of scale-free networks
  • Vertex correlation functions
  • Number of self-avoiding walks and circuits
  • Percolation transition
  • Critical phenomena of spin models defined on the static model
  • Conclusion

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

i static model of scale free networks

static model

I. Static model of scale-free networks

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide4

= adjacency matrix element(0,1)

static model

  • Degree of a vertex i:
  • Degree distribution:
  • We consider sparse, undirected, non-degenerate graphs only.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide5

static model

  • Grandcanonical ensemble of graphs:

 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)]

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide6

static model

  • When m=0  ER case.
  • Walker algorithm (+Robin Hood method) constructs networks in time O(N). N=107 network in 1 min on a PC.

Comments

  • Construction of the static model
  • Each site is given a weight (“fitness”)
  • In each unit time, select one vertex iwith prob. Pi and another vertex j with prob. Pj.
  • If i=j or aij=1 already, do nothing (fermionic constraint).Otherwise add a link, i.e., set aij=1.
  • Repeat steps 2,3 NK times (K = time = fugacity = L/N).

m = Zipf exponent

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide7

static model

  • Such algorithm realizes a “grandcanonical ensemble” of graphs G={aij} with weights

Each link is attached independently but with inhomegeous probability fi,j .

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide8

static model

Degree distribution

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide9

Recall

  • When l>3 (0<m<1/2),
  • When 2<l<3 (1/2<m<1)fij

1

fij2KNPiPj

3-l

Comments

static model

fij1

1

3-l

0

  • Strictly uncorrelated in links, but vertex correlation enters (for finite N) when 2<l<3 due to the “fermionic constraint” (no self-loops and no multiple edges) .

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

ii vertex correlation functions

Vertex correlations

II. Vertex correlation functions

Related work: Catanzaro and Pastor-Satoras, EPJ (2005)

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide11

Vertex correlations

Simulation results of k nn (k) and C(k)

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide12

Vertex correlations

Use this to approximate the first sum as

Our method of analytical evaluations:

For a monotone decreasing function F(x) ,

Similarly for the second sum.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide13

Vertex correlations

Result (1) for

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide14

Vertex correlations

Result (2) for

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide15

Vertex correlations

N

l

2

3

4

5

6

7

8

9

10

Result (3) Finite size effect of the clustering coefficient for 2<l<3:

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

iii number of self avoiding walks and circuits

Number of SAWs and circuits

III. Number of self-avoiding walks 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<l<3 ?

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide17

Number of SAWs and Circuits

The number of circuits or self-avoiding loops of size L on a graph is

with the first and the last nodes coinciding.

The number of L-step self-avoiding walks on a graph is

where the sum is over distinct ordered set of (L+1) vertices,

We consider finite L only.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide18

Number of SAWs and Circuits

and

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”.

Strategy for 2<l <3: For upper bounds, we use

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide19

Number of SAWs and Circuits

  • For , straightforward in the static model
  • For , the leading order terms in N are obtained.

Result(3): Number of L-step self-avoiding walks

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide20

Number of SAWs and Circuits

Typical configurations of SAWs (2<λ<3)

H

L=2

B

B

S

S

L=3

B

B

H

H

L=4

B

B

SS

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide21

Number of SAWs and Circuits

Results(4): Number of circuits of size

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

iv percolation transition

Percolation transition

IV. Percolation transition

Lee, Goh, Kahng and Kim, NPB (2004)

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide23

Percolation transition

  • The static model graph weight

can be represented by a Potts Hamiltonian,

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide24

Percolation transition

Partition function:

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide25

Percolation transition

Thermodynamic limit

  • Percolation transition at

l>4

l=4.8

3<l<4

l=3.6

2<l<3

l=2.4

Exact analytic evaluation of the Potts free energy:

  • Vector spin representation 
  • Integral representation of the partition function
  • Saddle-point analysis
  • Explicit evaluations of thermodynamic quantities.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

v critical phenomena of spin models defined on the static model

Spin models on SM

V. Critical phenomena of spin models defined on the static model

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide27

Spin models on SM

Spin models defined on the static model network can be analyzed by the replica method.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide28

Spin models on SM

When J i,j are also quenched random variables, extra averages on each J i,j should be done.

The effective Hamiltonian reduces to a mean-field type one with infinite number of order parameters,

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide29

Spin models on SM

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

We applied this formalism to the Ising spin-glass [DH Kim et al PRE (2005)]

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

slide30

Spin models on SM

To be compared with the ferromagnetic behavior for 2<l<3;

Critical behavior of the spin-glass order parameter in the replica symmetric solution:

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

vi conclusion

VI. Conclusion

1. 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 l, the degree exponent, is less than 3.

  • Two node and three node correlation functions, and the number of self-avoiding walks and circuits are obtained for 2<l <3. The walk statistics depend on the even-odd parity.

4. Kasteleyn construction of the Potts model is utilized to calculate thermodynamic quantities related to the percolation transition such as the mean number of independent loops.

5. The replica method is used to obtain the critical behavior of the spin-glass order parameters in the replica symmetry solution.

ICTP School and Workshop on Structure and Function of complex Networks (16-28 May 2005)

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