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

1. 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)

2. 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)

3. introduction I. Introduction Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

4. = 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)

5. 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)

6. static model II. Static model of scale-free networks Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

7. 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)

8. 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)

9. 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)

10. static model - Degree distribution - Percolation Transition Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

11. Recall When λ>3, When 2<λ<3 1 static model fijpNPiPj 3-λ fij1 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)

12. 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)

13. Other ensembles - Caldarelli et al, hidden variable model - Park-Newman Model Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

14. 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)

15. 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)

16. 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)

17. 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)

18. 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)

19. 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)

20. 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)

21. Spectral density V. Spectral density of adjacency and related matrices Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

22. 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)

23. 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)

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

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

26. Spectral density Similarly… Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

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

30. Spin models on SM VI. Ising spin-glass transitions Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006)

31. 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)

32. 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)

33. 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)

34. 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)

35. 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)