The 2nd KIAS Conference on Statistical Physics (NSPCS06), July 3-6, 2006, KIAS
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
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
2
Known so far
3
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
4
MF approach
for the Ising model in regular lattices
f
m*
m

droplet
5
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 ξ
 L. .
droplet
mN
- / ν
ψ(tN
1/ ν
).
-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
6
Generalization: FSS for the
mN
- / ν
 nTheory
ψ(tN
1/ ν
)
7
Conjecture:
FSS in SF networks with P(k) ~ k  γ
Note that our conjecture is independent of the type of network cutoffs!!
8
Langevin-type equation Approach
in SF networks
9
MF results in SF networks
10
Numerical Results

Extensive simulations are performed
on two different types of network cutoff
(Goh et al. PRL `01 for static; Cantanzaro et al. PRE `05 for UCM)


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.
11
Ising   6.50( 5)
mN
 /
  (tN
1 /
Static
UCM
Theory
(¼, 2)
(¼, 2)
Data
(0.25(1), 2.0(1))
(0.25(1), 2.0(1))
)
12
Ising
3 5
mN
 /
  (tN
1 /
)
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))
13
  2.75
CP on UCM:
UCM
Ours
C&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)
14
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 !!!
15
Unbounded density fluctuations of
CP on SF networks at criticality
16
Unbounded density fluctuations of CP:
Not only at criticality
But also everywhere on SF networks!!!
17