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Application of replica method to scale-free networks: Spectral density and spin-glass transition

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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)

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Outline

- Introduction
- Static model of scale-free networks
- Other ensembles
- Replica method – General formalism
- Spectral density of adjacency and related matrices
- Ising spin-glass transition
- Conclusion

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

introduction

I.Introduction

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

= adjacency matrix element (0,1)

introduction

- Degree of a vertex i:
- Degree distribution:

- We consider sparse, undirected, non-degenerate graphs only.

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

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

- 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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

static model

II.Static model of scale-free networks

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

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)

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

static model

- Construction of the static model

- Each site is given a weight (“fitness”)
- In each unit time, select one vertex i with 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 Np/2 times (p/2= time = fugacity = L/N).

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.

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

static model

- Such algorithm realizes a “grandcanonical ensemble” of graphs

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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

static model

- Degree distribution

- Percolation Transition

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Recall

When λ>3,

When 2<λ<3

1

static model

fijpNPiPj

3-λ

fij1

1

3-λ

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) .

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Other ensembles

III. Other ensembles

- Chung-Lu model

- Static model in this notation

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Other ensembles

- Caldarelli et al, hidden variable model

- Park-Newman Model

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

IV. 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.

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

- Since each bond is independently occupied, one can perform the graph ensemble average

The effective hamiltonian after the ensemble average is

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

- Under the sum over {i,j},

in most cases

- So, write the second term of the effective hamiltonian as

- One can prove that the remainder R is small in the thermodynamic limit. E.g. for the static model,

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

- The nonlinear interaction term is of the form

- So, the effective hamiltonian takes the form

- Linearize each quadratic term by introducing conjugate variables QR and employ the saddle point method

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

- The single site partition function is

- The effective “mean-field energy” function inside is determined self-consistently

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Replica method: General formalism

- 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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

V.Spectral density of adjacency and related matrices

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

is the ensemble average of density of states

with eigenvalues

for real symmetric N by N matrix M

It can be calculated from the formula

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

- Apply the previous formalism to the adjacency matrix

- Analytic treatment is possible in the dense graph limit.

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

Similarly…

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spectral density

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spin models on SM

VI.Ising spin-glass transitions

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

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

J i,j are also quenched random variables, do additional averages on each J i,j .

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spin models on SM

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

- 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.

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spin models on SM

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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

Spin models on SM

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

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

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

- 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.

VII. Conclusion

Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)