1. Finite-size scaling in complex networks Meesoon Ha (KIAS) in collaboration with Hyunsuk Hong (Chonbuk Nat’l Univ.) and Hyunggyu Park (KIAS) Korea Institute for Advanced Study The 2 nd KIAS Conference on Statistical Physics (NSPCS06), July 3-6, 2006, KIAS

2. 2 Outline Controversial issues of critical behavior in CP on scale-free (SF) networks MF vs. Non-MF? Network cutoff dependence? Mean-field (MF) approach and FSS for the Ising model in regular lattices FSS exponents in SF networks (from Ising to directed percolation, CP and SIS) Numerical results (with two different types of network cutoff) Summary

3. 3 Known so far

4. 4 Current controversial issues Non-MF Critical Behavior of Contact Process (CP) in SF networks? Castellano and Pastor-Satorras (PRL `06) claimed that the critical behavior of CP is non-MF in SF networks, based on the discrepancy between numerical results and their MF predictions. They pointed out the large density fluctuations at highly connected nodes as a possible origin for such a non-MF critical behavior. However, it turns out that all of their numerical results can be explained well by the proper MF treatment. In particular, the unbounded density fluctuations are not critical fluctuations, which are just due to the multiplicative nature of the noise in DP systems (Ha, Hong, and Park, cond-mat/0603787 ). (Ha, Hong, and Park, cond-mat/0603787 ). Cutoff dependence of FSS exponents? Natural cutoff vs. Forced sharp cutoff

5. 5 MF approach for the Ising model in regular lattices

6. 6 Why do we care this droplet length? For well-known equilibrium models and some nonequilibrium models, it is known that this thermodynamic droplet length scale competes with system size in high dimensions and governs FSS as. -Binder, Nauenberg, Privman, and Young, PRB (1985): 5D Ising model test - Luebeck and Jassen, PRE (2005): 5D DP model test -Botet, Jullien, and Pfeuty, PRL (1982): FSS in infinite systems

7. 7 Generalization: FSS for the Theory

8. 8 Conjecture: FSS in SF networks with Note that our conjecture is independent of the type of network cutoffs !!

9. 9 Langevin-type equation Approach in SF networks

10. 10 MF results in SF networks

11. 11  Extensive simulations are performed on two different types of network cutoff  Based on independent measurements two exponents and critical temperature are determined.  Our conjecture is perfectly confirmed well in terms of data collapse with our numerical finding. Numerical Results (Goh et al. PRL `01 for static; Cantanzaro et al. PRE `05 for UCM)

12. 12 Ising StaticUCM Theory (¼, 2) Data (0.25(1), 2.0(1))

13. 13 Ising Static (4.37)UCM (4.25) Theory(0.296, 2.46)(0.308, 2.60) Data(0.31(2), 2.46(10))(0.31(2), 2.75(20))

14. 14 CP on UCM: UCM OursC&P-S Theory(0.571, 2.33)(½, 2.67) Data(0.59(2), 2.44(10))(0.63(4), 2.4(2)) Ours (cond-mat/0603787) vs. Castellano and Pastor-Satorros (PRL ’06)

15. 15 Summary & Ongoing Issue  The heterogeneity-dependent MF theory is still valid in SF networks!  No cutoff dependence on critical behavior, if it is not too strong.  We conjecture the FSS exponent value for the Ising model and DP systems (CP, SIS), which is numerically confirmed perfectly well.  Heterogeneous FSS exponents for Synchronization? Thank you !!!

16. 16 Unbounded density fluctuations of CP on SF networks at criticality

17. 17 Unbounded density fluctuations of CP: Not only at criticality But also everywhere on SF networks!!!