Embed Size (px)
Citation preview
PHYSICAL REVIEW E, VOLUME 65, 066110
Ising model in small-world networks
Carlos P. HerreroInstituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientı´ficas (C.S.I.C.), Campus de Cantoblanco,
28049 Madrid, Spain~Received 21 January 2002; published 18 June 2002!
The Ising model in small-world networks generated from two- and three-dimensional regular lattices hasbeen studied. Monte Carlo simulations were carried out to characterize the ferromagnetic transition appearingin these systems. In the thermodynamic limit, the phase transition has a mean-field character for any finitevalue of the rewiring probabilityp, which measures the disorder strength of a given network. For small valuesof p, both the transition temperature and critical energy change withp as a power law. In the limitp→0, theheat capacity at the transition temperature diverges logarithmically in two-dimensional~2D! networks and as apower law in 3D.
DOI: 10.1103/PhysRevE.65.066110 PACS number~s!: 64.60.Cn, 64.60.Fr, 05.50.1q, 84.35.1i
anomksern
sicat
fgut-kn
ysi-
wsiseh
aveo
ep-t-tehe
asonin
nc-orld
sne-
el,any
et-
t-cial
hengomera-ge
lote-
the
op-
ec.a
arks
I. INTRODUCTION
Complex networks describe many systems in naturesociety, and have been modeled traditionally by randgraphs@1#. In the last years, models of complex networhave been introduced, motivated by empirical data in diffent fields@2#. In particular, small-world networks have beestudied, as they are suitable to describe properties of physystems with underlying networks ranging from regular ltices to random graphs, by changing a single parameter@3#.Watts and Strogatz@4# proposed a model for this kind onetwork, which is based on a locally highly connected relar lattice, in which a fractionp of the links between nearesneighbor sites are randomly replaced by new random linthus creating long-range ‘‘shortcuts.’’ The networks so geerated are suitable to study different kinds of physical stems, such as neural networks@5# and man-made communcation and transportation systems@4,6–8#.
These small-world networks interpolate between the tlimiting cases of regular lattices (p50) and random graph(p51). In the small-world regime, a local neighborhoodpreserved~as for regular lattices!, and at the same time somglobal properties of random graphs are maintained. Tsmall-world effect is usually measured by the scaling behior of the characteristic path length,, defined as the averagof the shortest distance between two sites. For a randnetwork one has a logarithmic increase of, with the networksizeN ~i.e., the number of sites!, while for a d-dimensionalregular lattice one expects an algebraic increase:,;N1/d. Incontrast, for a small-world network,, follows the scalinglaw @9–11#
,~N,p!;~N* !1/dF~N/N* !, ~1!
where the scaling functionF(u) has the limitsF(u);u1/d
for u!1 andF(u); ln u for u@1; N* ;p21 is a crossoversize that separates the large- and small-world regim@9,12,13#. This indicates that the small-world behavior apears for any finite value ofp(0,p,1) as soon as the nework is large enough, and in particular the global characistics of the network are changed dramatically in tpresence of only a small fraction of shortcuts.
1063-651X/2002/65~6!/066110~6!/$20.00 65 0661
d
-
al-
-
s,--
o
e-
m
s
r-
The importance of this shorter global length scale hbeen studied for several statistical physical problemssmall-world networks. Among these problems, one findsthe literature the signal propagation speed@4#, the spread ofinfections @14,15#, as well as site and bond percolatio@15–17#. The localization-delocalization transition of eletron states has been also studied on quantum small-wnetworks@18#.
Up to now, most of the published work on small worldhas concentrated on networks obtained from odimensional lattices. Barrat and Weigt@10# and Gitterman@19# have studied the crossover from one-dimensional~1D!to mean-field behavior for the ferromagnetic Ising modwhich presents a phase transition of mean-field type forvalue of the rewiring probabilityp.0, provided that thesystem size is large enough. Close top50 the transitiontemperatureTc goes to zero as 1/u log pu. A mean-field-typebehavior has also been found for theXY model in small-world networks generated from one-dimensional chains@20#.
More recently, Svenson and Johnston@21# have studiedthe damage spreading for Ising models on small-world nworks obtained by rewiring two-dimensional~2D! and three-dimensional~3D! regular lattices. They found that these neworks are more suitable than regular lattices to study sosystems with the Ising model.
Here we investigate the ferromagnetic transition for tIsing model in small-world networks generated by rewiri2D and 3D lattices. Contrary to the networks generated fr1D lattices, now one has phase transitions at finite tempturesTc.0 for p50. This means that one expects a chanfrom an Ising-type transition atp50 to a mean-field-typeone in the small-world regime. We employ Monte Car~MC! simulations to obtain average magnitudes for finisize systems. The resulting quantities are extrapolated tothermodynamic~infinite size! limit, where the small-worldbehavior is expected to dominate the thermodynamic prerties for any finite value of the rewiring probabilityp.0.
In Sec. II, we describe the computational method. In SIII, we present results of the MC simulations along withdiscussion. The paper closes with some concluding remin Sec. IV.
©2002 The American Physical Society10-1
eanb
eks
in
9%
iou
u-a
amth
aizrr
ueoa
te
1t
-th
e
bic-
s.
CARLOS P. HERRERO PHYSICAL REVIEW E 65 066110
II. COMPUTATIONAL METHOD
We consider the following Hamiltonian:
H52(i , j
Ji j SiSj , ~2!
whereSi561 (i 51, . . . ,N), and the coupling matrixJi j isgiven by
Ji j [H J~.0!, if i and j are connected,
0, otherwise.~3!
Monte Carlo simulations have been carried out on nworks of different sizes, generated from the 2D square3D cubic lattices. Small-world networks were generatedrandomly replacing a fractionp of the links of the regularlattices with new random connections. This procedure keconstant the total number of links in the rewired networThus, the average coordination numberz in the 2D and 3Dcases amounts to four and six, respectively. In the rewirprocess we avoided isolated sites~with zero links!. With thisprocedure we obtained networks in which more than 99.of the sites were connected in a single component~note thata random graph has usually many components of varsizes!. The remaining sites~when they appeared! were ex-cluded from the final networks employed for the MC simlations. The size of the networks used in our calculations wlarger than the crossover sizeN* @13,16#, so that we were inthe small-world regime.
The largest networks employed here included 2003200sites for the 2D system and 40340340 sites for the 3Dnetwork. Periodic boundary conditions were assumed. Spling of the configuration space has been carried out byMetropolis local update algorithm@22#. Several thermody-namic quantities and moments of the order parameter hbeen calculated for different simulation-cell sizes. Finite-sscaling was then employed to obtain the magnitudes cosponding to the thermodynamic limit~extrapolation to infi-nite size!. In the remainder of the paper, the presented valfor the different quantities will correspond to the extraplated ones, unless explicit mention is made indicating a pticular network size.
The ferromagnetic transition temperature has been demined by using Binder’s fourth-order cumulant@22#
UN~T
er
re
s
di-
to
rgdce
a
ili
In
o
nt
ll
ype
r d
-ats
ISING MODEL IN SMALL-WORLD NETWORKS PHYSICAL REVIEW E65 066110
a behavior different than that found in the 1D case, whTc;u log pu21. In the limit p51, one finds an increase inTc
as z rises from four~in 2D! to six ~in 3D!, as expected forrandom lattices, for which one haskBTc /zJ→1 asz→`.
To analyze the change in critical temperature withp, wecall DTc5Tc2Tc
0 , being Tc0 the transition temperature fo
the corresponding 2D or 3D regular lattice. In Fig. 3, wshow the dependence ofDTc uponp for the 2D and 3D casein a log-log plot. In both cases we find thatDTc can be fittedby a power-lawDTc;ps for p&0.01. The exponents is0.5260.03 for 2D and 0.9660.04 for 3D. Our result for 2Dnetworks is compatible with aAp dependence forDTc nearp50, which means that the derivativedTc /dp diverges as;1/Ap for p→0. In the 3D case, our results seem to incate a dependenceDTc;p for smallp. However, in this casethe lowestp values studied here may still be too highattain the small-p regime~see below!.
Associated with the increase inTc as the rewiring prob-ability p rises, one expects an increase in the critical eneE(Tc). We callec5E(Tc)/N the critical energy per site, anDec5ec2ec
0 its change with respect to the regular latti(p50). This differenceDec is shown in Fig. 4 as a functionof p for 2D ~squares! and 3D~circles! networks, in a log-logplot. Forp&0.01, Dec can be well fitted in both cases bypower law of the formDec;pu. For the exponentu, we findu50.4360.03 and 0.5660.04 in 2D and 3D, respectively.
A characterization of the ferromagnetic phase transitionthese networks requires the determination of the universaclass to which it corresponds. In the limitp50 ~regular lat-tices!, one has transitions of the 2D and 3D Ising type.order to determine the type of the phase transition atp.0,we have studied the critical exponentb, which gives thetemperature dependence of the order parameter close ttransition temperature:M &;(Tc2T)b for T,Tc . For thedifferent values of the rewiring probabilityp studied here, wehave calculated numerically the logarithmic derivative
FIG. 3. Dependence upon the rewiring probabilityp of the shiftin transition temperatureDTc with respect to the regular lattices, fo2D and 3D networks. Lines are guides to the eye.
06611
e
y
nty
the
m~ t !5d log^M &
d log t~6!
for t5Tc2T.0, which is related to the exponentb throughthe limit b5 limt→0m(t).
In Fig. 5, we present results for the derivativem as afunction of temperature for several values ofp and for a 2Dnetwork of size 2003200. For reference, we also preseresults of MC simulations forp50 ~Ising model on a regular2D lattice! for the same system size, which converge tob50.125, the critical exponent for the 2D Ising model. In acasesp.0, the extrapolationT→Tc gives an exponentbclose to 0.5, the value corresponding to a mean-field-t
FIG. 4. Dependence on the probabilityp of the shift in criticalenergy per siteDec with respect to the regular lattices, for 2D an3D networks. Lines are guides to the eye.
FIG. 5. Logarithmic derivativem versus the temperature differencet5Tc2T for small-world networks generated by rewiring2D lattice of size 2003200. Different symbols represent resulobtained for several values of the rewiring probabilityp. From topto bottom:p51, 0.1, 0.01, 0.001, and 0.
0-3
ms;
site
lu
teea
aE
ld
ass
an
as
t
ae
nn
io
iveef
w
rld
rkst
tion
y
dthe
theti-
all-g
C
lest
e
the
on
site
ent
an
Dr
ca-m-m-
CARLOS P. HERRERO PHYSICAL REVIEW E 65 066110
transition. However, for decreasingp, the observation of themean-field character of the transition requires to go to teperatures closer toTc ~or equivalently to larger system sizesee@22#!. Thus, forp50.001 we are still far from the value0.5 at the temperatures at which the employed systemallows us to give a precise value for the order parame^M &. But even in this case, the departure from the vaexpected for a 2D Ising-type transition is clear close toTc .
A complementary way to confirm the mean-field characof the phase transition consists in determining the exponn, which controls the behavior of the correlation length nethe critical temperature
j;uT2Tcu2n. ~7!
Close toTc , this critical exponent is related to the temperture dependence of the fourth-order cumulant defined in~4! as @20,22#
UN~T!'U* 1U1S 12T
TcDN1/n, ~8!
with U* and U1 independent ofT and the system sizeN.From this expression we have
DUN
DT}2N1/n, ~9!
which allows us to calculate the exponentn from the cumu-lant UN derived from the MC simulations. The values ofnobtained by this method agreed in all cases~within errorbars! with the critical exponent corresponding to mean-fietransitions:n50.5.
We conclude that the mean-field character of the phtransition ~which is the one found in random network!should appear in the thermodynamic limit for anyp.0. Thisis in line with the observation mentioned above that the chacteristics of the random graphs~e.g., mean distance betweesites l; logN) show up in small-world networks as soontheir sizeN is large enough.
To understand the dependence onp of the transition tem-peratureTc for small p, we will consider the two lengthscales present in this problem. On one side we havecorrelation lengthj, which nearTc follows the temperaturedependence given in Eq.~7!. On the other side, we havelength scale characteristic of the small-world network, givby the typical distance between ends of shortcuts:z5(pz)21/d @16#. When the correlation length is smaller thaz, the system behaves basically as a regular lattice. Whejgrows beyondz ~as happens when we approach the transittemperature from above,T→Tc
1), the ‘‘long-range’’ interac-tions introduced by the shortcuts come into play and grise to the mean-field behavior. Thus, the transition betwthe regular-lattice behavior and the mean-field one occursj'z. For the 1D Ising model, the correlation length at lotemperatures (kBT!J) is given by j;exp(2J/kBT) @24#.Taking into account that in this casez;1/p, the conditionj'z suggests a critical temperature for the 1D small-woTc;u log pu21, in agreement with more
06611
-
zer
e
rntr
-q.
e
r-
he
n
n
en
or
detailed calculations for this system@10#. For the 2DIsing model the correlation length nearTc scales asj;uT2Tc
0u21 @23,24#, whereas nowz;1/p1/2. Then, usingthe same argument, one expects for small-world netwogenerated from 2D lattices:Tc2Tc
0;p1/2, in good agreemenwith the results of our Monte Carlo simulations~see Fig. 3!.Thus, it seems that in general, for systems with a correlalength diverging asuT2Tc
0u2n, the order-disorder transitiontemperature for smallp depends on the rewiring probabilitas
Tc2Tc0;p1/nd. ~10!
Note that heren is the critical exponent of the consideremodel in the regular lattice, not the one corresponding tophase transition in the small-world network~which is themean-field one!. Hence, we see that the conditionj'z de-scribes the transition between the large-world regime~whichcorresponds to regular lattices! and the small-world one@13#,as well as the ferromagnetic transition, indicating thatsmall-world transition and the order-disorder one are inmately related.
The above argument, however, seems to fail for smworld networks built up from 3D lattices. For the 3D Isinmodel one has a critical exponentn'0.63 @24#, whichshould give for smallp a dependence:DTc;p0.53, with anexponent clearly smaller than that derived from our Msimulations, where we found forp&0.01: DTc;p0.96. Itseems that this discrepancy appears because the smalpvalue employed in our simulations (p51023) is still toolarge to observe the small-p behavior. Although for suchvalues ofp we are clearly in the small-world regime, thcorresponding value ofz in 3D (z;5), is too small to allowthe correlation length to reach the critical dependence ofIsing model in the regular lattice:j;uT2Tc
0u2n. In otherwords, to observe thep dependence ofTc given in Eq.~10!,one needsz values larger~i.e., smallerp values or largernetworks! than those employed here. A similar conclusiwas proposed for theXY model on 1D small-world networks@20#, for which MC simulations gave a dependenceTc'a log p1b ~with a and b numerical constants! instead ofTc;p, expected from the above argument.
For mean-field-type transitions, the heat capacity percv shows a finite jump~not a divergence! at Tc . Given thatat p50 one has Ising-type phase transitions with divergheat capacity atTc , such a jump incv will diverge in thelimit p→0. The temperature dependence ofcv is displayedin Fig. 6 for several values ofp and for networks built upfrom a 2003200 2D lattice. As expected, one observesincrease in the maximum value ofcv asp is reduced.
The maximum value ofcv for eachp, extrapolated to thethermodynamic limit, is shown in Fig. 7 for 2D and 3small-world networks. In Fig. 7~a!, we present our results fo2D, which for smallp display a logarithmic dependencecv
m
5a2b ln p, with the numerical constantsa51.7460.06 andb50.2060.01. This logarithmic dependence of the heatpacity can be explained by arguments similar to those eployed above to analyze the behavior of the transition te
0-4
s,
r
f
la
dien
amn
al
oc
sar
n
In
hation
A.
edy
l-
ob
d
ISING MODEL IN SMALL-WORLD NETWORKS PHYSICAL REVIEW E65 066110
perature asp→0. For the Ising model in 2D regular latticethe heat capacity nearTc
0 diverges as:cv(T);2 loguT2Tc0u
@23#. Taking into account the relation betweenTc and pgiven above for smallp, one finds for the 2D small-worldnetworkscv
m;2 log p, in agreement with the results of ouMonte Carlo simulations.
In the 3D Ising model,cv has atTc0 a singularity of the
form uT2Tc0u2a, with a'0.12 @24#. Assuming thatTc2Tc
0
;p1/nd, as in Eq.~10!, one expects forcvm close top50 a
power law of the form: cvm;pw, with an exponentw
52a/nd520.063. In Fig. 7~b!, we present the values ocv
m derived from our MC simulations as a function ofp in alog-log plot. These results are consistent with a power-dependencecv
m;pw at smallp. In fact, for p&0.01 our re-sults can be fitted with an exponentw520.05760.005.However, as in the case of the transition temperaturecussed above, smaller values ofp are necessary to determinunambiguously this dependence from numerical simulatioand in particular to find the exponentw.
IV. CONCLUDING REMARKS
We have studied the ferromagnetic transition that appefor the Ising model in small-world networks generated fro2D and 3D regular lattices. In these networks, the preseof a small disorder (p.0) causes a change in the university class of the order-disorder transition, from Ising forp50 to mean-field type forp.0.
Our results indicate that the order-disorder transitioncurs at a temperatureTc where the spin correlation lengthjis on the order of the lengthz ~typical distance between endof shortcuts!, characteristic of these networks. In particulclose top50 this gives a dependenceTc;u log pu21 in 1Dnetworks, andTc2Tc
0;p1/nd for networks generated fromregular lattices of higher dimensions. This is the depende
FIG. 6. Heat capacity per sitecv versus temperature for smalworld networks generated from a 2D lattice of size 2003200. Theplotted curves correspond to different values of the rewiring prability p, as indicated by the labels.
06611
w
s-
s,
rs
ce-
-
,
ce
found from our Monte Carlo simulations for 2D networks.the 3D case we find a power law forTc2Tc
0 , but the deter-mination of the actual exponent from MC simulations forp→0 requiresp values smaller~i.e., networks larger! thanthose employed here.
From the results presented in this paper, it is clear tthere is a close relation between the small-world transitand the order-disorder transition in this kind of network.
ACKNOWLEDGMENTS
The author benefited from useful discussions with M.R. de Cara and M. Saboya. Thanks are due to E. Chaco´n forcritically reading the manuscript. This work was supportby CICYT ~Spain! under Contract No. PB96-0874, and bDGESIC through Project No. 1FD97-1358.
-
FIG. 7. Maximum value of the heat capacity per sitecvm as a
function of the rewiring probabilityp. Symbols correspond to theextrapolation of finite-size results toN→`. ~a! 2D networks, in asemilogarithmic plot;~b! 3D networks, in a log-log plot. Dashelines are guides to the eye.
0-5
,
r-
-
A.
ro-
nd
l
n,
CARLOS P. HERRERO PHYSICAL REVIEW E 65 066110
@1# B. Bollobas, Random Graphs~Academic Press, New York1985!.
@2# S. H. Strogatz, Nature~London! 410, 268 ~2001!; R. Albertand A. L. Baraba´si, e-print cond-mat/0106096; S. N. Doogovtsev and J. F. F. Mendes, e-print cond-mat/0106144.
@3# D. J. Watts,Small Worlds~Princeton University Press, Princeton, 1999!.
@4# D. J. Watts and S. H. Strogatz, Nature~London! 393, 440~1998!.
@5# L. F. Lago-Ferna´ndez, R. Huerta, F. Corbacho, and J.Siguenza, Phys. Rev. Lett.84, 2758~2000!.
@6# V. Latora and M. Marchiori, Phys. Rev. Lett.87, 198 701~2001!.
@7# M. E. J. Newman, J. Stat. Phys.101, 819 ~2000!.@8# R. Albert, H. Jeong, and A. L. Baraba´si, Nature~London! 401,
130 ~1999!; A. L. Barabasi and R. Albert, Science286, 509~1999!.
@9# M. Barthelemy and L. A. N. Amaral, Phys. Rev. Lett.82, 3180~1999!; 82, 5180~1999!.
@10# A. Barrat and M. Weigt, Eur. Phys. J. B13, 547 ~2000!.@11# A. Scala, L.A. N. Amaral, and M. Barthe´lemy, Europhys. Lett.
55, 594 ~2001!.@12# A. Barrat, e-print cond-mat/9903323.
06611
@13# M. A. de Menezes, C. F. Moukarzel, and T. J. P. Penna, Euphys. Lett.50, 574 ~2000!.
@14# M. Kuperman and G. Abramson, Phys. Rev. Lett.86, 2909~2001!.
@15# C. Moore and M. E. J. Newman, Phys. Rev. E61, 5678~2000!.@16# M. E. J. Newman and D. J. Watts, Phys. Rev. E60, 7332
~1999!.@17# C. Moore and M. E. J. Newman, Phys. Rev. E62, 7059~2000!.@18# C. P. Zhu and S. J. Xiong, Phys. Rev. B62, 14 780~2000!; 63,
193 405~2001!.@19# M. Gitterman, J. Phys. A33, 8373~2000!.@20# B. J. Kim, H. Hong, P. Holme, G. S. Jeon, P. Minnhagen, a
M. Y. Choi, Phys. Rev. E64, 056 135~2001!.@21# P. Svenson and D. A. Johnston, Phys. Rev. E65, 036105
~2002!.@22# K. Binder and D. W. Heermann,Monte Carlo Simulation in
Statistical Physics, 3rd ed.~Springer-Verlag, Berlin, 1997!.@23# B. M. McCoy and T. T. Wu,The Two-Dimensional Ising Mode
~Harvard University Press, Cambridge, MA, 1973!.@24# J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newma
The Theory of Critical Phenomena~Oxford University Press,Oxford, 1992!.
0-6
