Reply to the Comment on the paper "Non-mean-field behavior of the contact process on scale-free networks"

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by  C. Castellano

1
Castellano and Pastor-Satorras Reply: The Com- uncorrelated networks [3], our original result α = 1/2 is
ment by Ha et al. [1] criticizes our recent result [2] that recovered. On the other hand, taking k ∼ N1/(γ−1),
c
the contactprocess (CP) on uncorrelatedscale-free(SF) whichcorrespondsto real correlated networks,yields the
networks does not behave according to heterogeneous value α = 1/(γ −1). This result, however, makes lit-
mean-field (MF) theory. The criticism is based on the tle sense,since the MFtheorydevelopedinourpaper[2]
following three claims: (1) The relative density fluctua- (andpresumablyin[1])dealswithuncorrelated networks.
7 tions discussed in Fig. 4 of [2] are well reproduced by a
0 Gaussian ansatz, Fig. 1 of [1]. (2) A numerical estimate
0 ofthedensitydecayforγ =2.75agreeswiththeMFpre-
In order to shed light on the true MF behavior of the
2 diction θ = 1/(γ−2). (3) An estimate of the finite-size
model, we haveperformed new simulations of the CP on
n scaling(FSS)exponentα=β/ν⊥ =0.59(2)forthesame
the random neighbors (RN) version of SF networks, in
a γ agreeswiththeMFconjectureα=1/(γ−1). Wereply
2 J to these three points in the following paragraphs. wthheicthwopcditffaekreesntthceute-xoaffctscMalFingvsalduiesc1u/ss2ed[2]a,bcoovnes.idInertinhge
(1)ThescalingformshowninFig. 1of[1]seemsrather
1 main plot of Fig. 1 we show results correspondingto the
interesting. However,it does not affect the main conclu-
estimate of the α exponent at p . We observe that the
] sions of our work. In fact, it proves that for almost all resultsfortherealisticcut-offk ∼c N1/2 deviatestrongly
n values of k, the unscaled relative density fluctuations r c
n k fromtheconjectureα=1/(γ−1),whichisnotevencor-
arelargerthan1andtheheightofthemaximumdiverges
- rectfortheunphysicalcut-off(inuncorrelatednetworks)
dis wthiethMtFheasnseutmwoprtkionsizteh.atAthdeivdeerngsiintgiesraρtkioarrekwceolnltdreafidnicetds tkoc ∼cheNck1/t(γh−e1v).alFuienoafllyt,hweeθceoxnpsoidneernta:msionrceenρa(ttu)r∼al tw−aθy,
. quantities, and therefore naturally hints more towards
t thefunctionρ(t)tθ shouldshowaplateauatintermediate
a failure of MF theory. Concerning the numerical points
m valuesoftforthecorrectvalueofθ. IntheinsetofFig.1
(2) and (3), we note that the critical point p = 0.4240
c the MF value θ = 1/(γ −2) is tested, for both cut-offs
- quoted in [1] is not compatible with our own estimate
d considered. As we can see, the MF value is reasonable
n pc =0.4215(5) [2], hinting that a diffent network model for large γ, but it fails completely for γ close to 2.
o is being used in [1]. In any case, the estimate of θ ex-
c tractedfromFig.2in[1]isutterlyimplausible: thevalue
[
isarbitrarilyobtainedfromanextremelyshortandnoisy

1 “plateau”, whose existence appears more as the effect of FromthisnumericalevidenceweconcludethattheMF
v chance than a real physical feature. Concerning point conjecture presented in the Comment is at best only ap-
5 (3), we are puzzled by the conjecture α = 1/(γ−1), of proximatelyvalidforthe unphysicalcaseofuncorrelated
7 which no proof is given, except a mention to “the hy- networks with cut-off k ∼ N1/(γ−1), which can only be
2 c
1 perscaling argument” [1]. We note, however, that it is constructed in the RN version of SF networks, while the
0 possible to obtain it from a trivial, although misleading, MF prediction for θ fails completely for small γ and any
7 argument: in Ref. [2] we observed that the MF density k . Therefore, the main conclusion of paper [2], the in-
c
0 of surviving particles at criticality should scale in uncor- validity of MF theory for real uncorrelated SF networks,
t/ related SF networks as ρ¯ ∼k−1, where k is the degree remains unchallenged.
a a c c
m cut-off. Assuming that kc ∼N1/2, corresponding to real
-
d
n 1 k = N1/2 Claudio Castellano1 and Romualdo Pastor-Satorras2
o 0.9 kcc = N1/(γ−1) 1012 1CNR-INFM, SMC, Dipartimento di Fisica,
Xiv:c α = β / ν⊥00..78 Comment Conjectur1/(γ−2)eρ(t) t 110006 UP2UD.nnleeiivvpAeearrlrssdtiiaott`aamMtdePonirtoRoldiot2em`e,cFaIn´-ıis0“ci0Lca1aad8i5SeEaRCpnaoigetminanazlyua,en”Irytiaaal,yNuclear
ar 0.6 100 101 t 102 103 Campus Nord B4, 08034 Barcelona,Spain
0.5
0.4
2 2.5 3 3.5 4 4.5 5
γ
[1] M.Ha,H.Hong,andH.Park,Phys.Rev.Lett.xx,xxxxxx
FIG. 1: Exponent α = β/ν⊥ for RN SF networks for both (2006), previous Comment.
cut-offscalings, compared withtheconjectureα=1/(γ−1).
[2] C. Castellano and R. Pastor-Satorras, Phys. Rev. Lett.
Inset: Check of the scaling ρ(t) ∼ t−1/(γ−2), for kc ∼ N1/2 96, 038701 (2006).
(dashed lines) and kc ∼ N1/(γ−1) (solid lines). Values of γ [3] M. Bogun˜´a, R. Pastor-Satorras, and A. Vespignani, Eur.
(bottom to top): 2.75, 2.50 and 2.20. Phys. J. B 38, 205 (2004).