1. KPS 2007 (April 19, 2007) On spectral density of scale-free networks Doochul Kim (Department of Physics and Astronomy, Seoul National University) Collaborators: Byungnam Kahng (SNU), Geoff. J. Rodgers (Brunel, UK)

2. KPS 2007 (April 19, 2007) Outline I. I.Introduction II. II.Matrices of Interest III. III.An Equilibrium Ensemble of Scale-Free Graphs – The Static Model IV. IV.Replica Method: General formalism V. V.Spectral Densities in the Dense Graph Limit VI. VI.Summary and Discussion

3. KPS 2007 (April 19, 2007) I.Introduction introduction Many real world networks are scale-free…

4. KPS 2007 (April 19, 2007) Internet is a network Nodes: Routers for IR network, Autonomous Systems for AS network Links: physical lines introduction

5. KPS 2007 (April 19, 2007) introduction Internet is scale-free Evolution of degree distribution of AS

6. KPS 2007 (April 19, 2007) introduction Internet is scale-free Load (or Betweenness Centrality) distribution of AS, AS+ and IR networks Load (or Betweenness Centrality) distribution of AS, AS+ and IR networks

7. KPS 2007 (April 19, 2007) http://www.nd.edu/~networks/gallery.htm There are many more examples that are scale-free approximately….. Information networks (WWW) Biological networks (Protein interaction network) Social network (Collaboration Network) introduction

8. KPS 2007 (April 19, 2007) introduction We consider sparse, undirected, simple graphs (no self-loops, no multiple bonds) with N nodes and L links (2L/N=p). Degree of a vertex i: SF degree distribution: = adjacency matrix with element 1 if connected and 0 otherwise introduction

9. KPS 2007 (April 19, 2007) introduction Spectral properties of matrices defined on such scale-free networks are of interest. For theoretical treatment, one needs to take averages of dynamic quantities over an ensemble of graphs. One can apply the replica method to obtain the spectral density of a class of scale-free networks, in the dense graph limit after the thermodynamic limit.

10. KPS 2007 (April 19, 2007) II.Matrices of Interest matrices of interest We consider 5 types of matrices associated with a graph G.

11. KPS 2007 (April 19, 2007) matrices of interest

12. KPS 2007 (April 19, 2007) matrices of interest

13. KPS 2007 (April 19, 2007) III.An Equilibrium Ensemble of Scale- Free Graphs – The Static model static model The static model is an efficient way of generating the scale-free network with arbitrary expected degree sequences.

14. KPS 2007 (April 19, 2007) 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 Chung-Lu model (2002) - -Closely related to the “hidden variable” models [Caldarelli et al PRL (2002), Boguna and Pastor- Satorras PRE (2003), Park-Newman (2003)]

15. KPS 2007 (April 19, 2007) static model 1. Each site is given a weight (“fitness”) 2. Select one vertex i with prob. P i and another vertex j with prob. P j. 3. If i=j or A ij =1 already, do nothing (fermionic constraint). Otherwise add a link, i.e., set A ij =1. 4. Repeat steps 2,3 Np/2 times ( =Np/2, p= fugacity for links).   Construction of the static model

16. KPS 2007 (April 19, 2007) static model   Such algorithm realizes a “grandcanonical ensemble” of graphs Each link is attached independently but with inhomegeous probability f i,j.

17. KPS 2007 (April 19, 2007) static model - Degree distribution - Percolation Transition

18. KPS 2007 (April 19, 2007) - c.f. Chung-Lu model: static model - Erdos-Renyi model :

19. KPS 2007 (April 19, 2007) IV. Replica method: General formalism Replica method: General formalism The replica method may be applied to perform the graph ensemble averages in the thermodynamic limit.

20. KPS 2007 (April 19, 2007) 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

21. KPS 2007 (April 19, 2007) 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

22. KPS 2007 (April 19, 2007) 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 rigorously that the remainder R/N is small in the thermodynamic limit for the equilibrium ensembles mentioned. E.g., for the static model, (PRE 2005)

23. KPS 2007 (April 19, 2007) 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

24. KPS 2007 (April 19, 2007) - The single site partition function is - The effective “mean-field energy” function inside is determined via the non-linear functional equation: Replica method: General formalism

25. KPS 2007 (April 19, 2007) - 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. Replica method: General formalism

26. KPS 2007 (April 19, 2007) V.Spectral Densities in the Dense Graph Limit Spectral density Formal expressions of the spectral density (the density of states) are obtained for various matrices. Explicit analytical results are obtained in the large p limit.

27. KPS 2007 (April 19, 2007) with eigenvalues is the ensemble average of density of states for real symmetric N by N matrix Q. It can be calculated from the formula Spectral density

28. KPS 2007 (April 19, 2007) Spectral density - Apply the previous formalism to the adjacency matrix - Analytic treatment is possible in the dense graph limit:

29. KPS 2007 (April 19, 2007) Spectral density

30. KPS 2007 (April 19, 2007) Spectral density

31. KPS 2007 (April 19, 2007) Spectral densitySimilarly…

32. KPS 2007 (April 19, 2007)

34. Spectral density

35. KPS 2007 (April 19, 2007)

36. Spectral density

37. KPS 2007 (April 19, 2007)

38. Spectral density

39. KPS 2007 (April 19, 2007)

42. VI. Summary and Discussion The replica method is formulated for a class of scale- free graph ensembles where each link is attached independently. General formula for the spectral densities of adjacency, Laplacian, random walk, weighted adjacency and weighted Laplacian matrices are obtained for sparse graphs (p=2L/N finite) in the thermodynamic limit. The spectral densities are obtained analytically in the large p limit. These results are expected to be a good approximation for 1 << p << N

43. KPS 2007 (April 19, 2007) The spectral densities at finite p, and/or finite N are unsolved problems except for special cases. The so-called eigenratio R for the weighted Laplacian can be estimated as ln R = |1-beta| ln N /(lambda-1). The Laplacian of the weighted network is a different problem that cannot be applied here. But several steps of approximations lead it to the weighted Laplacian treated in this work.