# structural correlations and critical phenomena of random scale-free networks:

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

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)

OutlineStatic model of scale-free networksVertex correlation functionsNumber of self-avoiding walks and circuitsPercolation transition Critical phenomena of spin models defined on the static modelConclusion

I.Static model of scale-free networksstatic model

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

static modelGrandcanonical 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)]

static modelEach 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 NK times (K = time = fugacity = L/N). Construction of the static model m = Zipf exponent

static modelSuch algorithm realizes a grandcanonical ensemble of graphs G={aij} with weightsEach link is attached independently but with inhomegeous probability f i,j .

static modelDegree distribution

static model

II. Vertex correlation functionsVertex correlationsRelated work: Catanzaro and Pastor-Satoras, EPJ (2005)

Vertex correlationsSimulation results of k nn (k) and C(k)

Vertex correlationsOur method of analytical evaluations: For a monotone decreasing function F(x) , Similarly for the second sum.

Vertex correlations

Vertex correlations

- Result (3) Finite size effect of the clustering coefficient for 2
- III.Number of self-avoiding walks and circuitsNumber of SAWs and circuitsThe 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
Number of SAWs and CircuitsThe number of L-step self-avoiding walks on a graph iswhere the sum is over distinct ordered set of (L+1) vertices,We consider finite L only.

- Number of SAWs and CircuitsStrategy for 2
Number of SAWs and CircuitsResult(3): Number of L-step self-avoiding walksFor , straightforward in the static modelFor , the leading order terms in N are obtained.

- Number of SAWs and CircuitsTypical configurations of SAWs (2
Number of SAWs and Circuits

IV. Percolation transitionPercolation transitionLee, Goh, Kahng and Kim, NPB (2004)

Percolation transitionThe static model graph weight can be represented by a Potts Hamiltonian,

Percolation transition

Thermodynamic limitExact analytic evaluation of the Potts free energy:Explicit evaluations of thermodynamic quantities.Vector spin representation Integral representation of the partition functionSaddle-point analysisPercolation transition

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

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

Spin models on SMThe effective Hamiltonian reduces to a mean-field type one with infinite number of order parameters,When J i,j are also quenched random variables, extra averages on each J i,j should be done.

Spin models on SMWe applied this formalism to the Ising spin-glass [DH Kim et al PRE (2005)]

Spin models on SMCritical behavior of the spin-glass order parameter in the replica symmetric solution:

- VI. Conclusion1.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
Static ModelN=31

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