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

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

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### I. Static model of scale-free networks

### II. Vertex correlation functions

### III. Number of self-avoiding walks and circuits

### IV. Percolation transition

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

### VI. Conclusion scale-free networks:

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)

Outline scale-free networks:

- 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)

static model scale-free networks:

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

= adjacency matrix element scale-free networks: (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)

static model scale-free networks:

- 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)

static model scale-free networks:

- 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)

static model scale-free networks:

- 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)

static model scale-free networks:

Degree distribution

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

- Recall scale-free networks:
- When l>3 (0<m<1/2),
- When 2<l<3 (1/2<m<1)fij

1

fij2KNPiPj

3-l

Comments

static model

fij1

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)

Vertex correlations scale-free networks:

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

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

Vertex correlations scale-free networks:

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

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

Vertex correlations scale-free networks:

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)

Vertex correlations scale-free networks:

Result (1) for

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

Vertex correlations scale-free networks:

Result (2) for

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

Vertex correlations scale-free networks:

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)

Number of SAWs and circuits scale-free networks:

- 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)

Number of SAWs and Circuits scale-free networks:

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)

Number of SAWs and Circuits scale-free networks:

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)

Number of SAWs and Circuits scale-free networks:

- 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)

Number of SAWs and Circuits scale-free networks:

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)

Number of SAWs and Circuits scale-free networks:

Results(4): Number of circuits of size

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

Percolation transition scale-free networks:

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

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

Percolation transition scale-free networks:

- 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)

Percolation transition scale-free networks:

Partition function:

’

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

Percolation transition scale-free networks:

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)

Spin models on SM scale-free networks:

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

Spin models on SM scale-free networks:

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)

Spin models on SM scale-free networks:

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)

Spin models on SM scale-free networks:

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)

Spin models on SM scale-free networks:

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)

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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