# 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 Presentation

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• 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