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

2. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) Outline I. I.Introduction II. II.Static model of scale-free networks III. III.Other ensembles IV. IV.Replica method – General formalism V. V.Spectral density of adjacency and related matrices VI. VI.Ising spin-glass transition VII. VII.Conclusion

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

4. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) introduction We consider sparse, undirected, non-degenerate graphs only. Degree of a vertex i: Degree distribution: = adjacency matrix element (0,1)

5. 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

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

7. 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)

8. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) static model 1. 1.Each site is given a weight (“fitness”) 2. 2.In each unit time, select one vertex i with prob. P i and another vertex j with prob. P j. 3. 3.If i=j or a ij =1 already, do nothing (fermionic constraint). Otherwise add a link, i.e., set a ij =1. 4. 4.Repeat steps 2,3 Np/2 times (p/2= time = fugacity =  L  /N).   Construction of the static model When λ is infinite  ER case (classical random graph). Walker algorithm (+Robin Hood method) constructs networks in time O(N).  N=10 7 network in 1 min on a PC.

9. 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 f i,j.

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

11. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) static model - 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). Recall When λ>3, When 2<λ<3 0 1 1 3-λ f ij  pNP i P j f ij  1

12. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) III. Other ensembles Other ensembles - Chung-Lu model - Static model in this notation

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

14. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) IV. Replica method: General formalism 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.

15. 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

16. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) Replica method: General formalism The effective hamiltonian after the ensemble average is - Since each bond is independently occupied, one can perform the graph ensemble average

17. 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,

18. 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 Q R and employ the saddle point method

19. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) - The single site partition function is - The effective “mean-field energy” function inside is determined self-consistently Replica method: General formalism

20. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) - 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 Replica method: General formalism

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

22. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) with eigenvalues is the ensemble average of density of states for real symmetric N by N matrix M It can be calculated from the formula Spectral density

23. 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.

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

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

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

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

29. Spectral density

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

31. 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.

32. 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.

33. 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

34. 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;

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

36. Workshop on Optimization in Complex Networks, CNLS, LANL (19-22 June 2006) Static Model N=3 1 Efficient method for selecting integers 1, 2, , N with probabilities P 1, P 2, , P N. Walker algorithm