Comment on "Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks"

0.12 MB·English

by  Meesoon Ha

1
Comment on “Non-Mean-Field Behavior of the -0.6
0.4230
Contact Process on Scale-Free Networks” -0.8 0.4233
-1 0.4237
ν|| 0.4240
Recently, Castellano and Pastor-Satorras [1] used the β/ -1.2 0.4250
- -1.4 -1.33
finite size scaling (FSS) theory to analyze simulation
-1.6
7 data for the contact process (CP) on scale-free networks
-1.8
0 (SFNs)andclaimedthatitsabsorbingcriticalbehavioris
0.001 0.01 0.1
0 notconsistentwiththemean-field(MF)prediction. Fur- 1/t
2
thermore, they pointed out large density fluctuations at

n highlyconnectedverticesasapossibleoriginfornon-MF FIG. 2: (Color online) Effective exponentsβ/ν for γ =2.75
||
a criticalbehavior. Inthis Comment, weproposeascaling with N =107. Allsamples are alivewhen dataare obtained.
J
theoryforrelativedensityfluctuationsinthespiritofthe
2
MF theory, which turns out to explain simulation data
1
perfectly well. We also measure the value of the critical expect rk ∼ (ρk)−1/2. With these two limiting results
] density decayexponent β/ν , which agreeswell with the and also with ρ ∼ N−α at criticality, one may assume a
h MFprediction. OurresultssktronglysupportthattheCP critical scaling relation as
c
e on SFNs still exhibits a MF-type critical behavior.
r =N(α−1/γ)/2f(kN−1/γ), (3)
m In contrast to equilibrium models with thermal noise, k
- nonequilibriummodelsexhibiting absorbingcriticalphe- where f(x) ∼ x(γ−1)/2 as x → 0 and f(x) ∼ x−1/2 as
at nomena involve so-called multiplicative noise. In partic- x → ∞. In Fig. 1, our simulation results at criticality
t ular, the noise amplitude is proportional to the square
s for γ =2.75 and 4.0 with various sizes N =103,··· ,106
. root of the particle density in models belonging to the
t showexcellentagreementswithourprediction. Notethat
a directed percolation class such as CP [2]. Therefore, at
m thecaseofγ =4.0wheretheγ-independentordinaryMF
the MF level neglecting spatial fluctuations, the particle
appears also shows the similar behavior in r .
- number fluctuation shouldbe proportionalto the square k
d We also observe the density decay dynamics for γ =
root of the total number of particles. It is easy to derive
n 2.75 in Fig. 2, where the effective decay exponents β/ν
o the relativedensity fluctuationonvertices ofdegreek as are plotted against 1/t. Our best estimate is β/ν =k
c k
[ rk =∆ρk/ρk ∼[NρkPk]−1/2, (1) 1.34(5)whichagreeswellwiththeMFpredictionofνk =
1 and β =1/(γ−2) for γ <3.

2 whereρk isthe partialdensity onverticesofdegreek, N Finally, we comment on the FSS exponent α = β/ν¯.
v
is the total number of vertices, and Pk is the degree dis- Vertices of degree k > N1/γ are practically irrelevant.
87 tribution. As ρk ∼ ρk near criticality (ρ: total density), Accordingly, the cutoff in k bigger than N1/γ can not
7 we find, for reasonably large k, influence critical FSS which should be determined by
3 the competition between correlated volume and system
r ∼(Nρ)−1/2k(γ−1)/2, (2)
0 k size. We conjecture that ν¯ = (γ −1)/(γ −2) from the
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where γ is the degree exponent. hyperscaling-type argument [3]. Our numerical estimate
0
/ For large enough k >N1/γ where the degree distribu- β/ν¯ = 0.59(2) for γ = 2.75 agrees well with our conjec-
at tion becomes almost flat in practice (NP ∼ O(1)), we ture. In fact, the results of Ref. [1] also agreereasonably
k
m well with our conjecture. In conclusion, we believe that
- the CP on SFNs exhibits the MF-type critical behavior.
d γ=2.75-103 This work was supported in part by the KRF Grant
n 104
o 105 -0.5 (MOEHRD) (R14-2002-059-01000-0).
c γ=4.0-110063 0.875 Meesoon Ha1,2, Hyunsuk Hong2, and Hyunggyu Park1
v: -yN 1 110045 1School of Physics, Korea Institutefor AdvancedStudy
Xi rk 106 -0.5 Seoul 130-722, Korea
1.5
r 2Department of Physics and RIPC
a Chonbuk National University,Jeonju 561-756, Korea
0.1
0.01 0.1 1 10 February 6, 2008
kN-c PACS numbers: 89.75.Hc, 05.70.Ln,87.23.Ge
FIG. 1: (Color online) Data collapse of the relative density [1] C. Castellano and R. Pastor-Satorras, Phys. Rev.
fluctuationsrk averagedoversurvivingsamplesinthesteady Lett. 96, 038701 (2006).
state for γ = 2.75 (pc = 0.4240, c = 0.363, y = 0.104) and [2] H. Hinrichsen, Adv. Phys. 49, 815 (2000).
γ =4.0 (pc =0.3570, c=0.25, y=0.125), respectively. [3] H. Hong, M. Ha, and H. Park (in preparation).