Introducing small-world network effects to critical dynamics

PHYSICAL REVIEW E 67, 026125 ~2003!

Introducing small-world network effects to critical dynamics

Jian-Yang Zhu1,2 and Han Zhu31CCAST (World Laboratory), Box 8730, Beijing 100080, China

2Department of Physics, Beijing Normal University, Beijing 100875, China*3Department of Physics, Nanjing University, Nanjing, 210093, China

~Received 22 July 2002; published 27 February 2003!

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model of twotypical small-world networks~SWN!, the adding type and the rewiring type. The general approaches and somebasic equations are systematically formulated. The rigorous investigation of the Glauber-type kinetic Gaussianmodel shows the mean-field-like global influence on the dynamic evolution of the individual spins. Accord-ingly a simplified method is presented and tested, which is believed to be a good choice for the mean-fieldtransition widely~in fact, without exception so far! observed for SWN. It yields the evolving equation of theKawasaki-type Gaussian model. In the one-dimensional Ising model, thep dependence of the critical point isanalytically obtained and the nonexistence of such a thresholdpc , for a finite-temperature transition, isconfirmed. The static critical exponentsg andb are in accordance with the results of the recent Monte Carlosimulations, and also with the mean-field critical behavior of the system. We also prove that the SWN effectdoes not change the dynamic critical exponentz52 for this model. The observed influence of the long-rangerandomness on the critical point indicates two obviously different hidden mechanisms.

DOI: 10.1103/PhysRevE.67.026125 PACS number~s!: 89.75.2k, 64.60.Ht, 64.60.Cn, 64.60.Fr

e

ut

ev

te

sa

relscfi-

llam-nd

giltssyigth

tingtheatic

heII

IV.va-sto

pesingm-m-

the

n-donein

call

eex-ac-

I. INTRODUCTION

Small-world networks~SWNs! are networks intermediatbetween a regular lattice and a random graph~see Refs.@1–4# and references therein for review!. They are believedto capture the essence of networks in reality, such as nenetworks@5#, power grids, social networks, and documenon the World Wide Web@6,7#, where remote vertices, whillocally clustered, often have the chance to be connectedshortcuts. Since 1998, when Watts and Strogatz presensimple model showing SWN effects@1#, it has been studiedintensively and extensively@8#. From the point of view ofstatistical physics, the presence of shortcuts assists thetem to behave as a whole, showing global coherenceSWN behavior.

One may be curious about the extent to which the featuof phase transitions will be different in spin-lattice modebuilt on small-world networks. Because the SWN effewidely exists in reality, this question is also of much signicance. Although new and interesting features have beenvealed recently@4,9–16#, it is still far from being completelyanswered. The dynamic aspect has been even less wederstood@17#. Naturally, we may expect the evolution ofsingle spin to be influenced partly by the overall systehowever, we find it difficult to offer more specific information. This is due to the complexity of the dynamics itself, athe often formidable mathematical task.

In this paper, we report our work on this interestinproblem—the critical dynamics of spin-lattice models buon SWN. As the introductory content, we shall first discuthe general approach. Then, among the various modeltems, we choose two special ones for a detailed investtion: the Gaussian model, which is relatively easy in ma

*Mailing address.

1063-651X/2003/67~2!/026125~12!/$20.00 67 0261

rals

iad a

ys-nd

s

t

re-

un-

,

ss-a--

ematics, and the one-dimensional Ising model. Intereskinetic features are revealed analytically and the study indynamic aspect also yields much information about the stproperties, such as the critical point.

This paper is organized as follows: In Sec. II we give tdefinition of spin-lattice models built on SWN. Section Icontains the discussion of the general approach~along with abrief review of the dynamic mechanisms!. The direct appli-cation to the kinetic Gaussian model can be found in Sec.In Sec. V, we present a simplified method and prove itslidity in Sec. V A by comparing its result with the rigorouone obtained in Sec. IV. Sections V B and V C are devotedthe further applications of this method on the Kawasaki-tyGaussian model and the one-dimensional Glauber-type Imodel, respectively. In Sec. VI, the influence of the randoness on the critical point is analyzed. Section VII is the sumarization with some discussions.

II. SPIN-LATTICE MODELS BUILT ON SMALL-WORLDNETWORKS

Following the first prototype@1# of SWN, there have beena number of variants~see, for example, Ref.@18#! in twobasic groups, which can be constructed as the following:initial network is, for example, a one-dimensional loop ofNvertices, each vertex being connected to its 2k nearest neigh-bors. ~1! Each pair of random vertices is additionally conected with probabilitypA ; ~2! the vertices are then visiteone after the other, and each link connecting a vertex toof its k nearest neighbors in the clockwise sense is leftplace with probability 12pR , and with probabilitypR it isreconnected to another randomly chosen vertex. We maythe first group adding-type~SWNs! and the second grouprewiring-type~SWNs!. Both of these modifications introduclong-range connections. The above algorithms can betended to systems with higher dimensionality and even fr

©2003 The American Physical Society25-1

inn

la-

f

i-

aguem

nxisdn

ar

enmK,

laelea

unls

ngIn

ed

-

aning

sm

-

m

-

J.-Y. ZHU AND H. ZHU PHYSICAL REVIEW E67, 026125 ~2003!

tal structures.~For example, for a spin located somewherea cubic lattice, three of the six bonds in some fixed directiowill be reconnected with probabilitypR , respectively.! Theseconstructions, with any~infinitesimally low! fraction ofshortcuts, allow us to reconcile local properties of a regunetwork ~clustering effect! with global properties of a random one~the average distancel; log10N).

A spin-lattice model built on SWN, which is the focus othis paper, can thus be defined: In aD-dimensional regularnetwork consisting ofN spins with periodic boundary condtion, each spin is linked to its 2kD nearest neighbors~in thispaper we will choosek51). Then~1! a certain number ofsupplemental links are added;~2! a portion of the bonds arerewired. Whether it is in group~1! ~adding-type SWN! orgroup ~2! ~rewiring-type SWN!, two spins connected byshortcut act in the same way as those connected by a rebond, that is, their interaction contributes to the systHamiltonian and they obey the redistribution~exchange!mechanism. In short, there is no difference between a lorange bond and a regular one. Other definitions may eFor example, since these small-world networks are mosystems for the networks in reality, one may have a reasochoose certain physical quantities, such asJ in the Isingmodel, to be different. However, although it is not necesswe will limit the scope of this paper to the above rule.

III. THE DYNAMIC MECHANISM

The various dynamic processes in the critical phenomare believed to be governed by two basic mechanisGlauber type with order parameter nonconserved andwasaki type with order parameter conserved. RecentlyGlauber-type single-spin transition mechanism@19,20# and aKawasaki-type spin-pair redistribution mechanism@21,22#have been presented as the natural generalizations of Ger’s flipping mechanism@23# and Kawasaki’s exchangmechanism@24#, respectively. They are generally applicaband mathematically well organized. Our work begins withreview of these two mechanisms. This is for the betterderstanding of the calculations in Secs. IV and V, and it amight be a convenient reference for further studies.

A. Single-spin transition mechanism

Glauber’s single-spin flipping mechanism allows an Isisystem to evolve with its spins flipping to their opposite.the single-spin transition mechanism@19#, a single spins i

may change itself to any possible values,s i , and the masterequation is

d

dtP~$s%,t !52(

i(s i

@Wi~s i→s i !P~$s%,t !

2Wi~ s i→s i !P~$s j Þ i%,s i ,t !#. ~1!

The transition probability is in a normalized form determinby a heat Boltzmann factor,

02612

s

r

lar

g-t.

elto

y,

as,a-a

ub-

-o

Wi~s i→s i !51

Qiexp@2bH~$s j Þ i%,s i !#,

Qi5(s i

exp@2bH~$s j Þ i%,s i !#, ~2!

whereb51/kBT. Based on the master equation, Eq.~1!, onecan prove that,

dqk~ t !

dt52qk~ t !1(

$s%F(

sk

skWk~sk→sk!GP~$s%;t !,

~3!

whereqk(t)[($s%skP($s%;t). There are also corresponding equations for the correlation functions.

B. Spin-pair redistribution mechanism

Kawasaki’s spin-pair exchange mechanism allowsIsing system to evolve with its nearest neighbors exchangtheir spin values. In the spin-pair redistribution mechani@21#, two connected spinss j and s l may change to anypossible values,s j and s l , as long as their sum are conserved. The master equation is

d

dtP~$s%,t !5(

j l &(

s j ,s l

@2Wjl ~s js l→s j s l !P~$s%;t !

1Wjl ~ s j s l→s js l !

3P~$s iÞ j ,s lÞk%,s j ,s l ;t !#. ~4!

The redistribution probability is also in a normalized fordetermined by a heat Boltzmann factor,

Wjl ~s js l→s j s l !51

Qjlds j 1s l ,s j 1s l

3exp@2bH~$sm%mÞ j ,l ,s j ,s l !#,

~5!

where the normalization factorQjl is

Qjl 5 (s j ,s l

ds j 1s l ,s j 1s lexp@2bH~$sm%mÞ j ,l ,s j ,s l !#.

Based on the master equation, Eq.~4!, one can prove that

dqk~ t !

dt522Dqk~ t !1(

$s%(w F (

sk ,sk1w

sk

3Wk,k1w~sksk1w→sksk1w!GP~$s%;t !, ~6!

whereD is the dimensionality and(w stands for the summation taken over the nearest neighbors.

5-2

bSomv

inea,ssefoim

eeree

utlie

s

-npe

th

ar

ion

fato

on

ia

hea

, n

nd

emse

toe

INTRODUCING SMALL-WORLD NETWORK EFFECTS TO . . . PHYSICAL REVIEW E67, 026125 ~2003!

C. How to apply them to SWN

In the construction of SWN, according to a certain proability, we will have a whole set of possible realizations.the theoretically correct way of treating dynamic systebuilt on SWN actually consists of three steps: First, we hato make a full list of all the possible realizations and poout the probability of each one of them. Second, we treach system respectively~apply the dynamic mechanismand obtain the master equation and the physical quantitieinterest!. Third, we give the expectation value with all theresults. This is cumbersome, but conceptually straightward. In Sec. V, we shall discuss whether there is any splified method.~In fact, there is.!

IV. KINETIC GAUSSIAN MODEL GOVERNED BY THEGLAUBER-TYPE MECHANISM

We have discussed the general approach right abovSec. III C, and in this section we will directly perform thcalculations according to the three steps, to study the thdimensional~3D! kinetic Gaussian model governed by thGlauber-type single-spin transition mechanism~see Sec.III A !. The calculation to be carried out is very long, bfortunately we can borrow some results from our earstudies@19,20#.

The Gaussian model, proposed by Berlin and Kac, icontinuous-spin model. Its Hamiltonian,

2bH5K(^ i , j &

s is j , ~7!

whereK5J/kBT. The spinssk can take any real value between (2`,1`), and the probability of finding a given spibetweensk andsk1dsk is assumed to be the Gaussian-tydistribution, f (sk)dsk; exp@2(b/2)sk

2#dsk , where b is adistribution constant independent of temperature. Thus,summation for the spin value turns into the integration(s

→*2`` f (s)ds. This model has been studied often as a st

ing point for investigations of other systems.Governed by the Glauber-type single-spin transit

mechanism, the expectation value of single spin@19# obeys

dqk~ t !

dt52qk~ t !1

K

b (^qk8 ,qk&

qk8~ t !. ~8!

Since in SWN, a spin may have a neighbor located veryfrom it, in the following calculations the summation isinclude every spin that is connected withsk . Taking averageof Eq. ~8!, we can obtain the evolution of the magnetizatiM (t)5(kqk(t)/N.

A. Kinetic Gaussian model on an adding-type small-worldnetwork

First, we treat the 3D kinetic Glauber-type Gaussmodel on an adding-type SWN consisting ofN ~a very largenumber! spins. The Kawasaki type is still tractable, but tmathematical task will be more complex. We will leave thtill Sec. V. Besides the regular bonds, each pair of spins

02612

-

sett

of

r--

in

e-

r

a

e

t-

r

n

to

matter how far apart, is connected via an additional bowith probability pA . Actually there are 2N(N21)/2 differentnetworks, each with a given probability. As a model systfor the networks in reality, we expect the number of thebonds,n;N(N21)pA/2, to be much smaller thanN. Sopractically we requirepAN!1. With respect to a specificspin s i jk , all the networks can be divided intoN groupslisted as the following.

~a! (0): There is no random bond ons i jk , and the prob-ability is (12pA)N21. According to Eq.~8!,

d

dtqi jk~ t !52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#.

~b! (1): There is only one random bond ons i jk , and theprobability isCN21

1 pA(12pA)N22. In fact, this group can befurther divided intoN21 subgroups, each correspondinga specific spin connected tos i jk , and each with the samprobability pA(12pA)N22. Averaging them, we get

dqi jk~ t !

dt52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#1K

bM ~ t !.

~c! (n): There aren random bonds ons i jk , and the prob-ability is CN21

n pAn(12pA)N2n21. Similarly, in average we

get,

dqi jk~ t !

dt52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#1nK

bM ~ t !.

~d! (N21): There areN21 random bonds ons i jk , andthe probability ispA

N21 .

dqi jk~ t !

dt52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#1K

b~N21!M ~ t !.

Thus, over all the realizations,

dqi jk~ t !

dt52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#1K

b (n50

N21

CN21n pA

n~1

2pA!N2n21nM. ~9!

Fortunately, we find the following relationship:

5-3

na

r

ree

h

1

t

th

the

eN,geiti-hy-

itsds

ill

theralow

any-ro-e

kallyed

ne-ave

oled,

lyhey


(n50

N21

CN21n pA

n~12pA!N2n21n5~N21!pA , ~10!

and thus

dqi jk~ t !

dt52qi jk~ t !1

K

b (w

@qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !#1K

b~N21!pAM ~ t !, ~11!

Similar results can be found in one- and two-dimensiomodels, and taking average we obtain

dM~ t !

dt52F12

2DK

b2

K

b~N21!pAGM ~ t !, ~12!

whereD is the dimensionality. The solution of Eq.~12! is

M ~ t !5M ~0!exp~2t/t!,

where the relaxation time

t51

12K/KcA

,

and the critical point

KcA5

b

2D1~N21!pA. ~13!

It is well known that for the critical point of the regulamodel,Kc

reg5b/2D, we can clearly see thatthe critical tem-perature will get higher as more long range bonds aadded. Actually, for a vertex located on an adding-typSWN, the number of the long-range bonds isnA;(N21)pA . In the small-world region, where the expectednAfor most of the vertices is very small, the change of tcritical point will be almost unperceivable.~The analysis ofthis result can be found in Sec. VI.!

B. Kinetic Gaussian model on a rewiring-type small-worldnetwork

Second, we treat the kinetic Gaussian model onrewiring-type SWN with a characteristic probabilitypR . Be-cause of the length, here we only give the details of theDcase. With regards to a specific spinsk , all the networks canbe divided into four major groups listed as the following,~1!both of the two regular bonds onsk , connectingsk21 andsk11, are not rewired;~2! the bond connectingsk21 andskis left unchanged but that connectingsk andsk11 is rewired;~3! the bond connectingsk andsk11 is left unchanged buthat connectingsk21 andsk is rewired;~4! both of them arerewired. The final result comes from the summation of allfour parts~see Appendix A for the details!:

02612

l

e

a

e

dqk~ t !

dt52qk~ t !1pR

2K

bM ~ t !

1~12pR!K

b@qk21~ t !1qk11~ t !#, ~14!

In two- and three-dimensional models, there are 24 and 26

major groups, respectively, and they can be treated insame way. Similar results are obtained:

dM~ t !

dt52S 12

2DK

b D M ~ t !. ~15!

Obviously, the relaxation time

t51

12K/KcR

,

where the critical point

KcR5Kc

reg5b/2D. ~16!

To summarize, we strictly obtain the evolution of thGaussian model built on SWN. On the adding-type SWthe critical temperature will get higher as more long ranbonds are added, while on the rewiring-type SWN the crcal temperature is unchanged. With the dynamic scalingpothesist;jz;uT2Tcu2zn, we havezn51, wherez is thedynamic critical exponent andn is the correlation lengthcritical exponent. We shall leave the summarization ofproperties to Sec. VII. The influence of the long-range bonon the critical point, which is a rather interesting topic, wbe discussed in Sec. VI.

The Gaussian model, being an idealization, often hasvalue of serving as a starting point for the more genestudies. The evolving equations of the individual spins shdistinctly the influence of the global coherence, whichauto-matically takes a mean-field-like form as if it comes fromaveraged spin.This character, as well as some alreadproved facts in earlier studies about the static behavior, pvides us with some helpful hints for the simplification of thmethod, which is the topic of the following section.

V. THE SIMPLIFIED METHOD

Presently, for the critical dynamics on SWN, we still laca well-established approach that should be both theoreticreliable and practically feasible. The above-mentionmethod~and the results! is theoretically rigorous, but it is toocomplex for the other models, even for the simplest odimensional Ising model. In the preceding section, we hpointed out that, inthe dynamic evolutionof the Gaussianmodel built on SWN, the influence of the system as a whon individual spins is mean-field-like. On the other hanrecent studies on the Ising model and theX-Y model built onSWN have also shown that the phase transition is ofthemean-fieldtype ~see Refs.@4,9–12,15# and Sec. V C for de-tails!. The various model systems built on SWN probabbelong to the same mean-field universality class, and t

5-4

thththiaaeeF

-il

g

ysekim

f

t

nal

qs.

g-

dinedin-

howedor-ldterb-

-N.s,

heerneddyav-

ibledy-us

,


might be treated with the same dynamic approach. Inproblem, which is found to be of the mean-field nature,following method may be the best choice. We deem allpossible networks as a single one. The effective Hamiltonof a spin-lattice model built on such a network is definedthe expectation value over all possible realizations. Its efftive behavior, e.g., the redistribution between two verticconnected with each other, is also the averaged result.example, two sites,s i ands j , are connected with probability p, then redistribution occurs between them with probabity pWi j . ~It is the basic assumption when we are treatinsystem governed by the Kawasaki-type mechanism.! Thenwe can directly apply the dynamic mechanisms to this stem. This method, which is of the mean-field nature, is ctainly more tractable in mathematics. We apply it to thenetic Gaussian model again and see if it will lead to the saresult.

A. Application in kinetic Gaussian model governed by theGlauber-type mechanism

In the 3D Gaussian model (N spins in total! built onadding-type SWN with periodic boundary condition, the efective Hamiltonian is

2bH5K(i , j ,k

s i jk~s i 11,j ,k1s i , j 11,k1s i , j ,k11!

11

2KpA(

i , j ,ks i jk (

i 8, j 8,k8s i 8 j 8k82

1

2KpA(

i , j ,ks i jk

2 .

~17!

With the Glauber-type transition mechanism, we beginderive the single-spin evolving equation according to Eq.~3!,along the same line of the calculations as in Ref.@19#. Weobtain

(s i jk

s i jkWi jk~s i jk→s i jk !5K

b F(w

~s i 1w, jk1s i j 1w,k

1s i j ,k1w!1~N21!pAsG ,~18!

where

s51

N21 S (lmn

s lmn2s i jk D'1

N (lmn

s lmn .

Substituting Eq.~18! into Eq. ~3!, we get

dqi jk~ t !

dt52qi jk~ t !1

K

b (w561

@qi 1w, jk~ t !1qi j 1w,k~ t !

1qi j ,k1w~ t !#1K

b~N21!pAM ~ t !. ~19!

02612

iseensc-sor

-a

-r--e

-

o

Similar results can be obtained for one- and two-dimensiomodels. These results, along with those forM (t), are exactlyin accordance with what we have obtained in Sec. III B, E~11! and ~12!.

In the 3D kinetic Gaussian model built on the rewirintype SWN, the effective Hamiltonian is,

2bH5K~12pR!(i , j ,k

s i jk~s i 11,jk1s i , j 11,k1s i , j ,k11!

1KpR

3

N (i , j ,k

s i jk (i 8, j 8,k8

s i 8 j 8k82KpR

3

N (i , j ,k

s i jk2 .

~20!

Similar calculations yield

dqi jk

dt52qi jk1pR

6K

bM1~12pR!

K

b (w

~qi 1w, j ,k

1qi , j 1w,k1qi , j ,k1w!.

This is also in accordance with the rigorous result, Eqs.~14!and~15!. Thus, in the kinetic Gaussian model, this simplifiemethod yields the same results as the rigorous ones obtawith the more complex standard method. As various splattice models, such as the Ising model, are believed to smean-field behavior on SWN, we believe that this simplifimethod is able to provide at least qualitatively correct infmation. In this sense, it is very different from the mean-fieapproximations taken in other universality classes. In lastudies, we will use it to study some more complex prolems; the following are two examples.

B. Application in kinetic Gaussian model governed by aKawasaki-type mechanism

Now we apply this simplified method to study the diffusion process in the kinetic Gaussian model built on SWAlthough we still have to deal with many complex equationit is relatively easy compared with the formidable task of tstandard approach. In such processes, the system is govby the Kawasaki-type redistribution mechanism. As alreamentioned at the beginning of this section, the system behior, just as the Hamiltonian, is also averaged over all possrealizations. The basic equations for the Kawasaki-typenamics listed below are generally applicable in varioorder-parameter-conserved processes.

~1! On D-dimensional adding-type SWN: Accordinglythe master equation should be modified as,

d

dtP~$s%,t !5(

j l &(

s j ,s l

@2Wjl ~s js l→s j s l !P~$s%;t !

1Wjl ~ s j s l→s js l !P~$s iÞ j ,s lÞk%,s j ,s l ;t !#

11

2pA(

j(lÞ j

(s j ,s l

@2Wjl ~s js l→s j s l !

3P~$s%;t !1Wjl ~ s j s l→s js l !

3P~$s iÞ j ,s lÞk%,s j ,s l ;t !#, ~21!

5-5

,

y,rention

pe

.


where the redistribution probabilityWjl (s js l→s j s l) is ofthe same form as Eq.~5!. With Eq. ~21!, we can get that

dqk~ t !

dt522Dqk~ t !1(

$s%(w F (

sk ,sk1w

sk

3Wk,k1w~sksk1w→sksk1w!GP~$s%;t !

1pAH 2~N21!qk~ t !1($s%

F(lÞk(

sk ,s l

sk

3Wkl~sks l→sks l !GP~$s%;t !J[Ak

(1)1pAAk(2) . ~22!

~2! On D-dimensional rewiring-type SWN: Accordinglythe master equation should be modified as

d

dtP~$s%,t !5~12pR!(

j l &(

s j ,s l


3P~$s%;t !1Wjl ~ s j s l→s js l !

3P~$s iÞ j ,s lÞk%,s j ,s l ;t !#1DpR

1

N21

3(j

(lÞ j

(s j ,s l


ic

-

02612

P~$s%;t !1Wjl ~ s j s l→s js l !

3P~$s iÞ j ,s lÞk%,s j ,s l ;t !#. ~23!

The redistribution probabilityWjl (s js l→s j s l) is of thesame form as Eq.~5!, and

dqk~ t !

dt5~12pR!H 22Dqk~ t !1(

$s%(w F (

sk ,sk1w

sk

3Wk,k1w~sksk1w→sksk1w!GP~$s%;t !J1

DpR

N21 H 2~N21!qk~ t !1($s} F(lÞk

(sk ,s l

sk

3Wkl~sks l→sks l !GP~$s%;t !J[~12pR!Rk

(1)1DpR

N21Rk

(2) . ~24!

AlthoughAk(1,2) andRk

(1,2) are of the same form, respectivelwe use different symbols because they are actually diffe~determined by the Hamiltonian-dependent redistributprobability W).

First we treat the Gaussian model built on the adding-tySWN. The first part,Ai jk

(1) , comes from the redistributionbetween the nearest neighbors. In the 3D case, with Eq~5!we can obtain

Ai jk(1)5

1

2~b1K !b$@~qi 11,j ,k2qi jk !2~qi jk2qi 21,j ,k!#1@~qi , j 11,k2qi jk !2~qi jk2qi , j 21,k!#1@~qi , j ,k112qi jk !

2~qi jk2qi , j ,k21!#%1K

2~b1K !@2~2qi 21,j ,k2qi 21,j 11,k2qi 21,j 21,k!1~2qi 21,j ,k2qi jk2qi 22,j ,k!

12~2qi 11,j ,k2qi 11,j 11,k2qi 11,j 21,k!1~2qi 11,j ,k2qi jk2qi 12,j ,k!12~2qi , j 21,k2qi , j 21,k112qi , j 21,k21!

1~2qi , j 21,k2qi jk2qi , j 22,k!12~2qi , j 11,k2qi , j 11,k112qi , j 11,k21!1~2qi , j 11,k2qi jk2qi , j 12,k!

12~2qi , j ,k212qi 11,j ,k212qi 21,j ,k21!1~2qi , j ,k212qi jk2qi , j ,k22!12~2qi , j ,k112qi 11,j ,k112qi 21,j ,k11!

1~2qi , j ,k112qi jk2qi , j ,k12!#. ~25!

Actually this is not as complex as it seems. With the lattconstanta, we can transform the above expression to be

Ai jk(1)5

3a2

b1K S b

62K D¹2q~r ,t !. ~26!

The second part,Ai jk(2) , comes from the redistribution be

tweens i jk and the farther spins,s i 8 j 8k8 . The result is

eAi jk

(2)52~N21!qi jk~ t !11

2 (i 8 j 8k8Þ i jk

@qi jk~ t !1qi 8 j 8k8~ t !#

1K

2b (i 8 j 8k8Þ i jk

(w

@„qi 1w, j ,k~ t !1qi , j 1w,k~ t !

1qi , j ,k1w~ t !…2„qi 81w, j 8,k8~ t !1qi 8, j 81w,k8~ t !

1qi 8, j 8,k81w~ t !…#, ~27!

5-6

ng

ns

exmtothac

n

pe

ee

a

thsy

t

l-vioe

za

ththbo

ing

ber

nsly

calrtsthe

on-e--

dical

htiesg-thehe

l-e

t,wned

forofg-

e

nue


Substituting Eqs.~26! and ~27! into Eq. ~22!, we obtain theevolving equation of the Gaussian model built on 3D additype SWN,

]q~r ,t !

]t5F 3a2

b1K S b

62K D1pA~N21!

a2K

2b G¹2q~r ,t !

1pA~N21!

2 S 126K

b D @M ~ t !2q~r ,t !#. ~28!

Similarly, on 3D rewiring-type SWN we obtain

]q~r ,t !

]t5H ~12pR!

3a2

b1K~12pR! Fb

62K~12pR!G

13pR

K

2b~12pR!J ¹2q~r ,t !1

3pR

2 F126K

b

3~12pR!G@M ~ t !2q~r ,t !#. ~29!

From these two equations we can clearly see the influeof the system built on SWN as a whole on individual spinOn regular lattices the evolution of the system can beplained by the diffusion mechanism, while on SWN, to sodegree, the individual spins will automatically adjust itselfapproach the average magnetization. This is obviouslyresult of the global coherence introduced by the small frtion of the long-range bonds.

On regular lattice, where the evolution is pure diffusio]q(r ,t)/]t5D¹2q(r ,t), and the diffusion coefficientD willvanish near the critical point. However, on adding-tySWN, where ]q(r ,t)/]t5D 8¹2q(r ,t)1C@M (t)2q(r ,t)#,two temperatures,Tc1

A andTc2A can be obtained by settingD 8

andC to zero, respectively.Tc1A will be lower than the critical

temperature on a regular lattice,Tcreg , and suggests that th

point at which the diffusion stops will be lowered by thrandomness.Tc2

A equalsTcreg , and at this point the evolution

will be pure diffusion. Similarly, there are also two tempertures for rewiring-type SWN, but we will haveTc1

R ,Tc2R

,Tcreg .

Obviously, the system behavior strongly depends ontemperature. For example, we study a one-dimensionaltem, and the initial magnetization isq(x,0)5sinx. When thetemperatureT.Tc2, both D 8 and C are positive, and themagnetization will approach homogeneity. WhenT,Tc1,both D 8 and C are negative, and the inhomogeneity gemore remarkable during the evolution. WhenTc2,T,Tc1 ,D 8.0 but C,0, and this is a more complex region. Athough here we can still easily predict the system behawith Eq. ~28! and obtain a stationary point, generally thevolution will be strongly dependent on the local magnetition.

C. Application to the Ising model

The second application of the simplified method is toone-dimensional ferromagnetic Ising model governed byGlauber-type mechanism. Recently, it has been studied

02612

-

ce.-

e

e-

,

-

es-

s

r

-

eeth

analytically@4,9# and with Monte Carlo simulations@10–14#.Due to the mathematical difficulties, the study was not govery smoothly at the beginning. In Ref.@9#, Gitterman con-cluded that the random long-range interactions, the numof which is above a minimal value, lead toa phase transi-tion. In Ref.@4#, Barrat and Weigt used some approximatioand expected a finite critical point ‘‘at least for sufficientlarge p and k>2’’ on rewiring-type SWN. They found outthat this transition is of the mean-field type. Although theycould not calculate the transition analytically, the numericomputation demonstrateda nonvanishing order parametein the presence of a vanishingly small fraction of shortcu,for k52 and 3. The numerical results seemed to supportfollowing relationship,Tc;22k/ log10pR . The more recentMonte Carlo studies have proved the above-mentioned cclusions on adding-type SWN and rewiring-type SWN, rspectively, but in those casesk51. The analysis of the relationship betweenTc and pR can be found in Ref.@11#. Thecritical exponents obtained from the simulation@11,12,15#,b'1/2, a'0, andn'1/2, further establish the mean-fielcharacter. However, although there is substantial numerproof, the nonexistence of such a thresholdpc , for a finite-temperature transition, is still to be confirmed.

In this paper, we will apply the simplified method, whicis of the mean-field nature, to study the dynamic properof the one-dimensional Ising model built on both addintype and rewiring-type SWN. We hope the success inGaussian model will continue in Ising system, of which tbehavior is already known to be the mean-field type. Athough we cannot obtain the full picture of the evolution, ware able to get the exactp dependence of the critical poinand some interesting critical exponents. As will be shobelow, our result agrees perfectly with the above-mentionnumerical simulation.

We find that the system shows very similar behavioradding- and rewiring-type SWN. We shall give the detailsrewiring-type SWN only, and report on the results of addintype SWN later.

For a rewiring-type SWN, the effective Hamiltonian is thsame as Eq.~20! ~the one-dimensional version!. We substi-tute it into the single-spin evolving equation, Eq.~3!, andobtain

dqk~ t !

dt52qk~ t !1(

$s%tanh@K~12pR!~sk211sk11!

12KpRs#P~$s%;t !, ~30!

wheres5( j Þks j /(N21)'( js j /N.If pR50, then it is the one-dimensional Ising model o

regular lattice, which we are familiar with. One can continto write

dqk~ t !

dt52qk~ t !1

1

2~qk211qk11!tanh 2K

and

5-7

or

eea

entic

bon

the

ba

rtion.o-

of

i-


dM~ t !

dt52M ~ t !~12tanh 2K !.

It yields M (t)}e2t/t, where t5(12tanh 2K)21. When K→Kc

reg5`, t;jz;(e2K)2, and thusz52.If the rewiring probabilitypR51, then

dqk~ t !

dt52qk~ t !1(

$s%tanh~2Ks !P~$s%;t !,

and

dM~ t !

dt52M ~ t !1(

$s%tanh~2Ks !P~$s%;t !.

Although ^ tanh(2Ks)&Þtanh(2KM), if we study the casewhen, near the critical point, the system is in almost though disorder, we will have^ tanh(2Ks)&; tanh(2KM);2KM, and

dM~ t !

dt;2M ~ t !1tanh@2KM ~ t !#.

This helps us to determine the critical pointKc51/2. WhenK,Kc51/2, then the system will be stable in a disorderstate withM50, but whenK>Kc51/2, there appears somkind of order. Taking Taylor expansion, one can find thwhenK is nearKc , M;(K2Kc)

1/2. This leads tob51/2 ~inthis specific situation!. We also get the relaxation timet;uK2Kcu21, with the scaling hypothesis~see below! wehavezn51.

Qualitatively similar situation should also be found whpR is between 0 and 1. First we assume that there is a critemperature, above which the system is disordered andlow which there begins to show nonzero magnetizatiFrom Eq.~30! we find that if initially M50 is given, thenthe system will stay in this disordered state. But belowcritical temperature this equilibrium will not be stable. Wcan determine the critical point introducing a small perturtion. WhenM→0,

dqk~ t !

dt'2qk~ t !1(

$s%tanh@K~12pR!~sk21

1sk11!#P~$s%;t !12KpR($s%

s$12tanh2@K~1

2pR!~sk211sk11!#%P~$s%;t !

52qk~ t !11

2~qk211qk11!tanh@2K~12pR!#

12KpR($s%

sH 121

2~11sk21sk11!tanh2@2K~1

2pR!#J P~$s%;t !. ~31!

02612

-

d

t

ale-.

e

-

BecauseM is very small, we believe 1/N (k^ssk21sk11& isan even smaller quantity of higher order. Thus

dM~ t !

dt52M1M tanh@2K~12pR!#

12KpRH 121

2tanh2@2K~12pR!#J M . ~32!

The critical point can be determined as

tanh@2KcR~12pR!#12Kc

RpRH 121

2tanh2@2Kc

R~12pR!#J51. ~33!

If K,KcR , the disordered stateM50 will be stable, but if

K.KcR , a small perturbation will drive the system apa

from the disordered state towards nonzero magnetizatNow we continue to find several interesting critical expnents.

~1! x;uT2Tcu2g: Near the critical point,Meq→0. If aweak fieldH is introduced, we will have

2bH5(k

skFK~12pR!sk111pR

K

N21 (j Þk

s j1H

kBTG ,and

dqk~ t !

dt52qk~ t !1(

$s%tanhFK~12pR!~sk211sk11!

12KpRs1H

kBTGP~$s%;t !. ~34!

Following the same way as that taken in the calculationthe critical point ~in fact, we can just replace 2KpRs by2KpRs1H/kBT), we get

d

dtM ~ t !52M ~ t !1M ~ t !tanh@2K~12pR!#1S 2KpRM ~ t !

1H

kBTD H 121

2tanh2@2K~12pR!#J . ~35!

From Eq.~35!, one can easily find that in a system in equlibrium near the critical point,K5Kc

R1D, and

x[]M

]H;D21. ~36!

Thus,g51.~2! M;uT2Tcub: WhenM→0, we take

1

N (k

^ssk21sk11&;M3.

Then from Eq.~31! we can get

5-8

o-

-e

be

ate

seni-

ltsanding

r a

ell

ge

ofch

on

of

sde-ys-seay-N,

hlso

ser-n

e

red

e


dM~ t !

dt'2S 12tanh@2K~12pR!#22KpR

3H 121

2tanh2@2K~12pR!#J D M ~ t !

2KpRtanh2@2K~12pR!#M3~ t !.

When K2KcR5D→0, let dM(t)/dt50, and one will get

M2;D by taking Taylor expansion. Thus, the critical expnentb51/2.

~3! t;jz;uT2Tcu2zn: When studying the critical slowing down, we can assumeM to be very small, and thus wuse Eq.~32!. It yields

M ~ t !5M ~0!e2t/t,

where

t21512tanh@2K~12pR!#22KpR

3H 121

2tanh2@2K~12pR!#J .

If K→KcR , thent→`. If K5Kc

R2D, andD→0, then onewill find t;D21. Thuszn51. It has been found in MonteCarlo simulations thatn'1/2, so z52. It is of the samevalue as that obtained on the regular lattice.

1D Ising model on adding-type SWN showqualitativelythe samebehavior Its critical point can be determined as

tanh 2KcA1Kc

A~N21!pAS 121

2tanh22Kc

AD51, ~37!

and we have foundthe same critical exponents, g, b, andz.For a vertex on an adding-type SWN lattice, the num

of the long-range bonds isnA;(N21)pA , while for a ver-tex on a rewiring-type SWN lattice,nR;2pR . Then depen-dence of critical pointKc of 1D Ising model on adding-typeand rewiring-type SWN, Eqs.~33! and~37!, can be found in

FIG. 1. Then dependence of critical pointKc of 1D Ising modelon adding-type and rewiring-type SWN.

02612

r

Fig. 1. One can clearly see the above mentioned approximrelationship, Kc

A}2 log10nA;2 log10@(N21)pA#, or KcR}

2 log10nR;2 log10(2pR).When nA and nR are small enough~in the small-world

region!, from Eqs.~33! and ~37! we can get

nR.~12tanh 2KcR!Y S 1

2Kc

Rtanh22KcRD ,

nA.~12tanh 2KcA!Y FKc

AS 121

2tanh22Kc

AD G .When nA and nR are approaching zero,Kc

A,R→`. For thesame value of the critical point, (nR2nA)→01. As shown inFig. 1, for most of the region, the two curves are very cloto each other. However, though the difference may be infitesimal, the curve ofnR is always above that ofnA , as isdistinct when these curves are relatively large.

VI. THE INFLUENCE OF THE RANDOMNESS ON THECRITICAL POINT

Regarding the behavior of the critical point, our resushow interesting contrast between the Gaussian modelthe Ising model. Here, we shall mention another interestmodel system, theD-dimensional mean-field~MF! Isingmodel, which might help us understand this problem. Forandomly selected spin in this model, each of its 2D nearestneighbors is replaced by an averaged one, and it is wknown that the critical point isKc51/2D. A simple calcula-tion will yield that on rewiring-type SWN,Kc

R will notchange, while on adding-type SWN, since the long-ranbonds increase the contact of a spin with the system,Kc

A

51/(2D1nA).Our result of the 1D Ising model is in contrast to that

the MF Ising model and the Gaussian model, while for eaone of them the critical temperature will be very closeadding-type SWN and rewiring-type SWN~the former willbe higher! in the small-world region. This may be a resultthe totally different role played by the long-range bonds.~1a!In the D-dimensional MF Ising model, the critical point isolely determined by the mean coordination number thatcides the coupling between an individual spin and the stem. It is unchanged on rewiring-type SWN but will increaon adding-type SWN. Thus, the critical temperature will stunaltered on rewiring-type SWN but will be slightly increased by the long-range bonds on adding-type SWwhich typically take up only a small fraction.~1b! As isshown by earlier studies@19#, the Gaussian model, thougbeing a very different system, has a critical point that aonly depends on the mean coordination number. On SWN,its critical point is very similar to the MF Ising model. It ibelieved to be mainly a mathematical result, and to undstand this we shall review the calculations in Sec. IV. Orewiring-type SWN, for an individual spin, the long ranginteraction partly replaces the nearest-neighbor~nn! cou-pling. However, the coordination number can be consideunchanged, since no bonds are created or eliminated~theyare just redirected!. As a result, on the right-hand side of th

5-9

oua

ua,illnbee-n

ar

het

wn

Ncronq

ghheembram

ps

-hooe

ol

ht

A

-hete

ctheaian

int

g

g-ns.areng-arVI.

b-

nd

ofentThe

n of

d in

twoo-ioninga

.ntsheion

ianot

Fem

there-can

-icsthin

hisit isp-nu-

t is

mayhch


evolving equations of the spins, the lost part of the nn cpling is exactly compensated by the MF term. The criticpoint is determined by taking average of the evolving eqtions of the spins, which only consist of linear terms. Thusis mathematically straightforward that the critical point wstay the same. On adding-type SWN, the consideratiosimilar, except that here the coordination number willslightly increased.~2! The crossover observed in the ondimensional Ising model is certainly governed by a differemechanism. For example, on rewiring-type SWN, theretwo competing length scales: the correlation lengthj; exp(2J/kBT), and the characteristic length scale of tSWN, which can be taken as the typical distance betweenends of a shortcut@25#, z;pR

21/D5pR21 . When j,z, the

system basically behaves as a regular lattice, otherwise itshow MF behavior as an effect of the long range interactioThe transition occurs atj'z, suggesting a critical pointTc;u log10pRu21. Since the two structures, rewiring-type SWand adding-type SWN, have the same length scales, thecal point can hardly be separated in the small-world regiHowever, interestingly we find that the mechanism in E~1!, i.e., long-range bonds→ coordination number~interac-tion energy! → critical point, can also be observed, thoubeing a minor factor. As is mentioned in Sec. V C, for tsame expected number of long-range bonds, the critical tperature on adding-type SWN will be higher than that otained on rewiring-type SWN. We expect it to be a genephenomenon, though not yet reported by the numerical silations.

VII. SUMMARY AND DISCUSSION

In this paper, we study the critical dynamics, Glauber tyand Kawasaki type, on two typical small-world networkadding type and rewiring type.

The logical sequence.As the introductory content we discuss the general approach of the critical dynamics, whictheoretically straightforward but may be mathematically tcomplex. We directly apply it to the kinetic Gaussian modgoverned by Glauber-type mechanism and obtain its evtion. We observe that, in thedynamicevolution of the indi-vidual spins, the influence of the system as a whole, whicthe result of the presence of the long-range bonds, takesmean-field-like form as if it comes from an averaged spin.the same time, earlier studies have revealed the~MF! staticbehavior of the Ising model andX-Y model. These both suggest us to present the following simplified method. All tSWN realizations are deemed as a single one, with botheffective Hamiltonian and the effective behavior averagover all of them. It is tested in the same model and exaleads to the same rigorous result. Then this method, whicbelieved to be theoretically reliable and mathematically fsible, is applied to two more difficult problems, the Gaussmodel governed by Kawasaki-type mechanism and the odimensional kinetic Ising model.

The Gaussian model.~1! Whether it is built on adding-type SWN or rewiring-type SWN, the long-range bondstroduce the mean-field-like global influence of the systemthe dynamic evolution of the individual spins. On addin

02612

-l-

it

is

te

he

ills.

iti-..

--lu-

e,

is

lu-

ishet

hedlyis-

ne-

-o-

type SWN such influence is additional while on rewirintype SWN, it partly replaces that of the neighboring spi~2! On adding-type SWN, as more long-range bondsadded, the critical temperature gets higher; but on rewiritype SWN it does not differ at all from that of the regullattice. This interesting discrepancy is explained in Sec.~3! In both networks, the relaxation timet51/(12K/Kc),and thuszn51. This dynamic property has also been otained on regular lattices and fractal lattices@19–22#. It ishighly universal, independent of the geometric structure athe dynamic mechanism.~4! On SWN, the evolution of theKawasaki-type model can be viewed as the combinationtwo mechanisms, the diffusion and the automatic adjustmof the single spin to approach the average magnetization.pure diffusion equation (]/]t)q(r ,t)5D¹2q(r ,t) will bemodified as (]/]t)q(r ,t)5D 8¹2q(r ,t)1C@M (t)2q(r ,t)#.By setting D 8 and C to zero we will get two competingcharacteristic temperatures, instead of the single definitiothe critical point for the regular lattices (D50). The tem-perature dependence of the dynamic evolution is discusseSec. V B.

The Ising modelanalytically studied by the simplifiedmethod: The system shows very similar behavior on thenetworks, adding-type SWN and rewiring-type SWN. Intrducing a very small perturbation of the local magnetizatto the disordered state, we obtain the critical point by judgthe stability of the equilibrium. The inexistence of suchthresholdpc for a finite-temperature transition is confirmed1

From the dynamic equation, we obtain the critical exponeg andb in agreement with the numerical simulation and talready-proved MF behavior of the system. The relaxattime is divergent near the critical point ast;uT2Tcu21, andthus zn51 ~note the same relationship in the Gaussmodel!. In the 1D Ising model, the SWN effect does nchange the dynamic critical exponentz52.

The influence of the randomness on the critical point.Ourresult of the 1D Ising model is in contrast to that of the MIsing model and the Gaussian model. For each one of ththe critical temperature on adding-type SWN,Tc

A , will behigher than that on rewiring-type SWN,Tc

R , obtained for thesame expected number of the long-range bonds, thoughdifference may be hardly perceivable in the small-worldgion. A detailed analysis of the responsible mechanismsbe found in Sec. VI.

Prospects.We hope that further studies on critical dynamics will continue to reveal interesting dynamic characteristof the widely existing critical phenomena combined wiSWN. The simplified method shall become a useful toolthis field. As the phase transition is of the MF nature, tmethod is certainly the best choice, and in this sensebasically different from the custom MF approximation aplied to other universality classes. Presently, besides themerical study, analytical treatment in the dynamic aspec

1As mentioned above, for some physical considerations onedefine a differentJ8 for the long-range bonds, which might be mucsmaller thanJ. With this method we can prove that there is not sua thresholdJc8 either.

5-10

alsela

c

(1b-

d

heond


scarce compared to the study of the static properties@4,9#. Infact, at least in some cases, such a study may be feasibleuseful indeed. Our work, especially on the Ising model, ashows that the study of the dynamic aspect is often ablyield much information on the general properties in a retively convenient way.

ACKNOWLEDGMENT

This work was supported by the National Natural ScienFoundation of China under Grant No. 10075025.

APPENDIX A: 3D GAUSSIAN MODEL OF THEREWIRING-TYPE SWN

~1!. The two regular bonds onsk ~connectingsk21 andsk11) have not been rewired, and the probability is2p)2. This group can be further divided into many sugroups:

~1a! There are no random bonds onsk . The probability is(12p)2(12@1/(N21)p#N22,

dqk

dt52qk1

K

b~qk211qk11!.

~1b! There is only one random bond onsk . The probabilityis

~12p!2CN221 1

N21pS 12

1

N21pD N23

,

and

dqk

dt52qk1

K

b~qk211qk111M !.

~1c! There aren random bonds onsk , and the probability is

~12p!2CN22n S 1

N21pD nS 12

1

N21pD N2n22

.

This group can be further divided intoCN21n subgroups, each

corresponding ton specific spins connected tosk via therandom bonds, and with the same probability

~12p!2S 1

N21pD nS 12

1

N21pD N2n22

.

In the i th subgroup,

dqk

dt52qk1

K

b S qk211qk111(l 51

n

qi l D .

Thus, averaging them we get

02612

ndoto-

e

dqk

dt52qk1

K

b~qk211qk111nM!.

~1d! There areN22 random bonds onsk , and the prob-ability is (12p)2@1/(N21)p#N22,

dqk

dt52qk1

K

b S qk211qk111 (l 51

N22

qi l D52qk1

K

b@qk211qk111~N22!M #.

Thus, the first part of the time derivative of single spin, is

S dqk

dt D1

5~12p!2F2qk1K

b~qk211qk11!

1K

b (n50

N22

CN22n S p

N21D n

3S 121

N21pD N2n22

nMG . ~A1!

~2! The bond connectingsk21 andsk is left unchanged,but that connectingsk andsk11 has been rewired. This boncan be rewired to each one of theN21 spins with equalprobability (12p)p/N. In each case, we can analyze tsituation in the way described above. For example, the bis rewired tos j .

~2a! There are no random bonds onsk . The probability is

~12p!p

N~12p!2S 12

1

N21pD N22

.

Also,

dqk

dt52qk1

K

b~qk211qj !.

~2b! There is only one random bond onsk . The probabil-ity is

~12p!p

NCN22

1 1

N21pS 12

1

N21pD N23

.

Also,

dqk

dt52qk1

K

b~qk211qj1M !.

~2c! There aren random bonds onsk , and the probabilityis

~12p!p

NCN22

n S 1

N21pD nS 12

1

N21pD N2n22

.

Also

5-11

i

’’

e

the


dqk

dt52qk1

K

b~qk211qj1nM!.

~2d! There areN22 random bonds onsk , and the prob-ability is @1/(N21)p#N22. Also

d

dtqk52qk1

K

b~qk211qj1~N22!M !.

Thus the second part of the time derivative of single spin

S dqk

dt D2

5~12p!pF2qk1K

b~qk211M !

1K

b (n50

N22

CN22n S p

N21D n

3S 121

N21pD N2n22

nMG . ~A2!

~3! The bond connectingsk andsk11 is left unchanged,but that connectingsk21 andsk has been rewired~we omitthe very small probability that this bond may be ‘‘rewiredto sk). Now there is only one regular bond onsk . @Payattention that this case is different from Eq.~2!.# Based on asimilar consideration, we have the third part of the time drivative of single spin which obeys

-

et

.A

wsy,

02612

s

-

S dqk

dt D3

5~12p!pF2qk1K

bqk111

K

b (n50

N22

CN22n S p

N21D n

3S 121

N21pD N2n22

nMG . ~A3!

~4! Both the bonds connectingsksk11 andsk21sk havebeen rewired. Based on similar consideration, we havefourth part of the time derivative of single spin,

S dqk

dt D4

5p2F2qk1K

bM1

K

b (n50

N21

CN22n S p

N21D n

3S 121

N21pD N2n22

nMG . ~A4!

Applying Eq. ~10!, we get

dqk

dt5(

i 51

4 S dqk

dt Di

52qk1p2K

bM1~12p!

K

b~qk211qk11!.

~A5!

t

t

tter

@1# D.J. Watts and S.H. Strogatz, Nature~London! 393, 440~1998!.

@2# D.J. Watts,Small Worlds~Princeton University Press, Princeton, 1999!.

@3# M.E.J. Newman, C. Moore, and D.J. Watts, Phys. Rev. L84, 3201~2000!.

@4# A. Barrat and M. Weigt, Eur. Phys. J. B13, 547 ~2000!.@5# L.F. Lago-Fernandez, R. Huerta, F. Corbacho, and J

Siguenza, Phys. Rev. Lett.84, 2758~2000!.@6# R. Albert, H. Jeong, and A.L. Barabasi, Nature~London! 401,

130 ~1999!; A.L. Barabasi and R. Albert, Science286, 509~1999!.

@7# M.E.J. Newman, Phys. Rev. E64, 016131~2001!; 64, 016132~2001!.

@8# From the great amount of literature appearing recently,only give several examples. Signal propagation speed andchronizability in Ref. @1#; M. Kuperman and G. AbramsonPhys. Rev. Lett.86, 2909~2001!; C. Moore and M.E.J. New-man, Phys. Rev. E61, 5678 ~2000!; percolation in Ref.@25#;D.S. Callawayet al., Phys. Rev. Lett.85, 5468 ~2000!; C.P.Zhu and S.J. Xiong, Phys. Rev. B63, 193 405~2001!.

@9# M. Gitterman, J. Phys. A33, 8373~2000!.@10# Andrzej Pekalski, Phys. Rev. E64, 057104~2001!.

t.

.

en-

@11# Carlos P. Herrero, Phys. Rev. E65, 066110~2002!.@12# H. Hong, Beom Jun Kim, and M.Y. Choi, e-prin

cond-mat/0204357.@13# H. Hong, Beom Jun Kim, and M.Y. Choi, e-prin

cond-mat/0205608.@14# P. Svenson and D.A. Johnston, Phys. Rev. E65, 036105

~2002!.@15# Beom Jun Kim, H. Hong, Petter Holme, Gun Sang Jeon, Pe

Minnhagen, and M.Y. Choi, Phys. Rev. E64, 056135~2001!.@16# H. Hong, M.Y. Choi, and Beom Jun Kim, Phys. Rev. E65,

047104~2002!.@17# S. Jespersen, I.M. Sokolov, and A. Blumen, Phys. Rev. E62,

4405 ~2000!.@18# S. Jespersen and A. Blumen, Phys. Rev. E62, 6270~2000!; V.

Karimipour and A. Ramzanpour,ibid. 65, 036122~2002!.@19# Jian-Yang Zhu and Z.R. Yang, Phys. Rev. E59, 1551~1999!.@20# Jian-Yang Zhu and Z.R. Yang, Phys. Rev. E61, 210 ~2000!;

61, 6219~2000!.@21# Han Zhu and Jian-Yang Zhu, Phys. Rev. E66, 017102~2002!.@22# Han Zhu, Jian-Yang Zhu, and Yang Zhou, Phys. Rev. E66,

036106~2002!.@23# R.J. Glauber, J. Math. Phys.4, 294 ~1963!.@24# K. Kawasaki, Phys. Rev.145, 224 ~1966!.@25# M.E.J. Newman and D.J. Watts, Phys. Rev. E60, 7332~1999!.

5-12

Documents

Introducing small-world network effects to critical dynamics