Finite-size scaling Finite-size scaling in complex networksin complex networks

Meesoon Ha (KIAS)Meesoon Ha (KIAS)

in collaboration within collaboration with

Hyunsuk Hong (Chonbuk Nat’l Univ.) and Hyunggyu Park (KIAS)Hyunsuk Hong (Chonbuk Nat’l Univ.) and Hyunggyu Park (KIAS)

Korea Institute for Advanced Study

The 2nd KIAS Conference on Statistical Physics (NSPCS06), July 3-6, 2006, KIAS

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

Known so far

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).

Cutoff dependence of FSS exponents? Natural cutoff vs. Forced sharp cutoff

5

MF approach for the Ising model in regular lattices

m

f

*m

droplet

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 .

.ξdroplet L

-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

).ψ(tNNm ν1/ν/-

7

)ψ(tNNm ν1/ν/-

Generalization: FSS for the Theory n

8

Conjecture: FSS in SF networks with γk~P(k)

Note that our conjecture is independent of the type of network cutoffs!!

9

Langevin-type equation Approach in SF networks

10

MF results in SF networks

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

Ising )( /1/ tNmN

Static UCM

Theory (¼, 2) (¼, 2)

Data (0.25(1), 2.0(1)) (0.25(1), 2.0(1))

)5(50.6

13

Ising )( /1/ tNmN

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))

53

14

CP on UCM:

UCMOurs C&P-S

Theory (0.571, 2.33) (½, 2.67)

Data (0.59(2), 2.44(10)) (0.63(4), 2.4(2))

75.2

Ours (cond-mat/0603787) vs. Castellano and Pastor-Satorros (PRL ’06)

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

Unbounded density fluctuations of CP on SF networks at criticality

17

Unbounded density fluctuations of CP: Not only at criticalityBut also everywhere on SF networks!!!