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Vertex Correlations, Self-Avoiding Walks and Critical Phenomena on the Static Model of

Vertex Correlations, Self-Avoiding Walks and Critical Phenomena on the Static Model of Scale-Free Networks. DOOCHUL KIM (Seoul National University). Collaborators:

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Vertex Correlations, Self-Avoiding Walks and Critical Phenomena on the Static Model of

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  1. Vertex Correlations, Self-Avoiding Walks and Critical Phenomena on the Static Model of Scale-Free Networks 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) International Workshop on Complex Networks, Seoul (23-24 June 2005)

  2. Outline • Static model of scale-free networks • Vertex correlation functions • Number of self-avoiding walks and circuits • Critical phenomena of spin models defined on the static model • Conclusion International Workshop on Complex Networks, Seoul (23-24 June 2005)

  3. static model I. Static model of scale-free networks International Workshop on Complex Networks, Seoul (23-24 June 2005)

  4. = adjacency matrix element(0,1) static model • Degree of a vertex i: • Degree distribution: • We consider sparse, undirected, non-degenerate graphs only. International Workshop on Complex Networks, Seoul (23-24 June 2005)

  5. static model • For theoretical treatment, one needs to take averages over an ensemble of graphs G.  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)] International Workshop on Complex Networks, Seoul (23-24 June 2005)

  6. 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 International Workshop on Complex Networks, Seoul (23-24 June 2005)

  7. 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 . International Workshop on Complex Networks, Seoul (23-24 June 2005)

  8. static model Degree distribution Percolation Transition International Workshop on Complex Networks, Seoul (23-24 June 2005)

  9. 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) . International Workshop on Complex Networks, Seoul (23-24 June 2005)

  10. Vertex correlations II. Vertex correlation functions Related work: Catanzaro and Pastor-Satoras, EPJ (2005) International Workshop on Complex Networks, Seoul (23-24 June 2005)

  11. Vertex correlations Simulation results of k nn (k) and C(k) on Static Model International Workshop on Complex Networks, Seoul (23-24 June 2005)

  12. 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. International Workshop on Complex Networks, Seoul (23-24 June 2005)

  13. Vertex correlations Result (1) for International Workshop on Complex Networks, Seoul (23-24 June 2005)

  14. Vertex correlations Result (2) for International Workshop on Complex Networks, Seoul (23-24 June 2005)

  15. 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: International Workshop on Complex Networks, Seoul (23-24 June 2005)

  16. 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 ? International Workshop on Complex Networks, Seoul (23-24 June 2005)

  17. 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. International Workshop on Complex Networks, Seoul (23-24 June 2005)

  18. 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 International Workshop on Complex Networks, Seoul (23-24 June 2005)

  19. 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 International Workshop on Complex Networks, Seoul (23-24 June 2005)

  20. 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 International Workshop on Complex Networks, Seoul (23-24 June 2005)

  21. Number of SAWs and Circuits Results(4): Number of circuits of size International Workshop on Complex Networks, Seoul (23-24 June 2005)

  22. Spin models on SM IV. Critical phenomena of spin models defined on the static model International Workshop on Complex Networks, Seoul (23-24 June 2005)

  23. Spin models on SM Spin models defined on the static model network can be analyzed by the replica method. International Workshop on Complex Networks, Seoul (23-24 June 2005)

  24. Spin models on SM When J i,j are also quenched random variables, do additional averages on each J i,j . The effective Hamiltonian reduces to a mean-field type one with infinite number of order parameters, International Workshop on Complex Networks, Seoul (23-24 June 2005)

  25. 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)] International Workshop on Complex Networks, Seoul (23-24 June 2005)

  26. 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: International Workshop on Complex Networks, Seoul (23-24 June 2005)

  27. V. 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. The replica method is used to obtain the critical behavior of the spin-glass order parameters in the replica symmetry solution. International Workshop on Complex Networks, Seoul (23-24 June 2005)

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