Statistical mechanics of topological phase transitions in networks

0.38 MB·English

by  Gergely Palla

Statistical mechanics oftopologicalphasetransitions innetworks
Gergely Palla, Imre Dere´nyi, Ille´s Farkas, and Tama´s Vicsek
Biological Physics Research Group of HAS and Department of Biological
Physics, Eo¨tvo¨s University, Pa´zma´ny P. stny. 1A, H-1117 Budapest, Hungary
(February2,2008)
4
0 Weprovideaphenomenologicaltheoryfortopologicaltransitionsinrestructuringnetworks.Inthisstatistical
0
mechanicalapproachenergyisassignedtothedifferentnetworktopologiesandtemperatureisusedasaquantity
2
referringtothelevelofnoiseduringtherewiringoftheedges.Theassociatedmicroscopicdynamicssatisfiesthe

n detailedbalanceconditionandisequivalenttoalatticegasmodelontheedge-dualgraphofafullyconnected
a network. In our studies – based on an exact enumeration method, Monte-Carlo simulations, and theoretical
J considerations–wefindarichvarietyoftopologicalphasetransitionswhenthetemperatureisvaried. These

0 transitionssignalsingularchangesintheessentialfeaturesoftheglobalstructureofthenetwork. Depending
3 ontheenergyfunctionchosen,theobservedtransitionscanbebestmonitoredusingtheorderparametersΦs =
smax/M,i.e.,thesizeofthelargestconnectedcomponentdividedbythenumberofedges,orΦk =kmax/M,

] thelargestdegreeinthenetworkdividedbythenumber ofedges. If,forexampletheenergyischosentobe
h
E = smax, theobserved transitionisanalogous tothepercolationphase transitionof randomgraphs. For
c −
thischoiceoftheenergy,thephase-diagraminthe[ k ,T]planeisconstructed. Singlevertexenergiesofthe
e h i
m formE = if(ki),wherekiisthedegreeofvertexi,arealsostudied.Dependingontheformoff(ki),first
orderandcontinuousphasetransitionscanbeobserved. Incaseoff(ki)= (ki+α)ln(ki),thetransitionis
- continuous,Pandatthecriticaltemperaturescale-freegraphscanberecovere−d. Finally,byabruptlydecreasing
t
a thetemperature,non-equilibriumprocesses(e.g.,nucleation,growthofparticulartopologicalphases)canalso
t beinterpretedbythepresentapproach.
s
.
t
a PACSnumbers:9.75.Hc,05.70.Fh,64.60.Cn,87.23.Ge
m
-
d I.INTRODUCTION randomgraphmodel[5,6],whichoccursbyvaryingtheaver-
n agedegree, k ,oftheverticesaround k = 1. For k < 1
o the graphfahllsiapartinto small pieces,honithe other hhanid for
c Inrecentyears,theanalysisofthenetworkstructureofin-
k 1agiantconnectedcomponentemerges(inadditionto
[ teractionshas becomea popularand fruitfulmethod used in h i≥
the finite components). Another subtle example is the map-
thestudyofcomplexsystems.Whenevermanysimilarobjects
2 ping of a growing network model onto an equilibrium Bose
v in mutual interactions are encountered, these objects can be
gas[7]. Forthelattermodel,undercertainconditionsasingle
6 representedasnodesandtheinteractionsaslinksbetweenthe
nodeisallowedtocollectafinitefractionofalledges,corre-
5 nodes,defininganetwork. Theworld-wide-web,thescience
5 spondingtoahighlypopulatedgroundlevelandsparselypop-
citation index, and biochemical reaction pathways in living
9 ulatedhigherenergiesseeninBose-Einsteincondensation.
cellsareallgoodexamplesofcomplexsystemswidelymod-
0
3 eled with networks, and the set of further phenomenawhere Whenconnectingthegraphtheoreticalaspectsofnetworks
0 the network approach can be used is even more diverse. In tostatisticalphysics, onecanstep furtherfromthe analogies
/ mostcases, theoverallstructureofnetworksreflectthechar- bydirectlydefiningstatisticalensemblesforgraphs. Theuse
t
a acteristicpropertiesoftheoriginalsystems,andenableoneto ofastatisticalmechanicalformalismforthechangesingraphs
m
sortseeminglyverydifferentsystemsintoafewmajorclasses beinginanequilibrium-likestateisexpectedtoprovideasig-
- ofstochasticgraphs[1, 2]. Thesedevelopmentshavegreatly nificantlydeeperinsightintotheprocessestakingplaceinsys-
d
n advancedthe potentialto interpretthe fundamentalcommon temsbeinginasaturatedstateand,assuch,dominatedbythe
o featuresofsuchdiversesystemsassocialgroups,technolog- fluctuatingrearrangementsoflinksbetweentheirunits.
c ical, biological and other networks. The effects of both the
: As an example, let us take a given number of units inter-
v restructuring[3] and the growth[4] of the associated graphs
acting in a “noisy” environment. These units can be people,
i have been considered, leading to a number of exciting dis-
X firms, genes, etc. The probability for establishing a new or
coveriesaboutthe lawsconcerningtheirdiameter,clustering
r ceasinganexistinginteractionbetweentwounitsdependson
a anddegreedistribution. Realnetworkstypicallyexhibitboth
boththenoiseandtheadvantagegained(orlost)whenadopt-
aspects (growth and rearrangement)one of which is usually
ingthenewconfiguration. Inthispicture,aglobaltransition
dominating the dynamics. Here we concentrate on the evo-
in the connectivity properties can occur as a function of the
lutionofgraphsduetorestructuring,butshallbrieflydiscuss
level of noise. For instance, if the conditions are such that
thegrowthregimeaswell.
the interactions between the partners become more “conser-
Variousinterestingeffectsobservedinnetworkscanbein- vative” (safer choices are more highly valued), then – as we
terpreted using analogies with well understood phenomena showlater–atransitionfromalessorderedtoamoreordered
studiedinstatisticalphysics.Asaclassicalexample,wemen- networkconfigurationcantakeplace.Inparticular,ithasbeen
tionthepercolationphasetransitionintheErdo¨s-Re´nyi(ER) argued[8]thatdependingonthelevelofcertaintypesofun-
2
certainties (expected fluctuations) business networks reorga- framefortheoptimizationofnetworkstructure(forexamples
nize from a star-like topologyto a system of morecohesive, of network optimization problemssee [18, 19]). To find the
highlyclusteredties. optimal configuration for a given task, the system has to be
Thereareseveralpossiblewaystodefinethestatisticalen- cooled,usinganappropriateenergyfunction.
semble of networks. In [9, 10], the members of the ensem- Thisarticleisadirectextensionofourpreviouswork[20];
blewereidentifiedbytheFeynmandiagramsofafieldtheory coveringmoredetails, results, andnew approaches. The pa-
in zero dimensions (called “minifield”), and the weights of perisorganizedasfollows:inSec. IIwedefinethecanonical
the graphs were given by the corresponding amplitudes cal- ensembleofthenetworkstogetherwiththepartitionfunction
culatedusingthestandardFeynmanrules. Theensembleob- andotheressentialthermodynamicquantities. InSec. IIIwe
tained this way was characterizedby the fractaland spectral discuss the numericalmethods used to study the phase tran-
dimensions,andthedependenceofthetopologyofthegraphs sitions. InSec. IV wepresentthephasetransitionsobtained
on these two parameterswas discussed. The authors argued for energiesthat dependon global properties, and Sec. V is
that in the parameter plane of two parameters related to the devotedtotwointerestingcasesofsinglevertexenergies. In
fractal and spectral dimension, the region of generic graphs Sec. VI we discuss briefly the grand canonical ensemble of
and the region of crumpled graphs are separated by a line; networksandweconcludeinSec. VII.
and on this separating line scale-free networks appear. An
alternative definition for the partition function was proposed
byBergandLa¨ssigin[11],resultinginasimplerformalism, II.STATISTICALMECHANICSOFNETWORKS
analogous to the statistical mechanics of classical Hamilto-
nian systems. They introduced a Hamiltonian for networks,
We shall consider a set, g , of undirected graphs, con-
a
andalsoaparameterβplayingtheroleofinversetemperature. taining N nodes and M lin{ks.}Each graph g can be repre-
a
Theweightsofdifferentgraphsinthepartitionfunctionwere sentedbytheadjacencymatrixAa,whereAa =1ifvertices
ij ij
obtained from these two quantities as in classical statistical
iandj areconnectedanditiszerootherwise. Inaheatbath
mechanics. These studies showed that Hamiltoniansbeyond
attemperatureT,thecanonicalensembleofthesegraphs(in
thesinglevertexform(wheretermsdependingonconnectivi- analogywiththatproposedin[11]),canbedefinedbythepar-
tiesbetweentheverticesalsoappear)leadtocorrelationsbe-
titionfunction
tween the vertices for large β. A similar model leading to
interesting results was presented in [12], where the Hamil- Z(T)= e−Ea/T, (1)
tonian depended on the ratios of the degrees of neighboring
{Xga}
vertices,andthedynamicsfavoreddisassortativemixingand
highclustering. Thesystemorganizeditselfintothreephases where E is the energy assigned to the different configura-
a
dependingonone parameter: the exponential,scale-freeand tions.
hub-leavesstates were produced, respectively. However, the The restructuring processes of the network can be inter-
non-uniform selection of links at the rewiring in this model preted via the following physical picture: The basic event
makesitimpossibletosatisfythedetailedbalancecondition. of rearrangement is the reallocation of a randomly selected
In this paper we analyze the reorganization of networks edge (link) to a new position either by “diffusion” (keeping
from the pointof view of topologicalphase transitions, i.e., one end of the edge fixed and connectingthe other one with
transitions in the graph structure as a function of tempera- a new node) or by removing the given edge and connecting
ture, the quantity representing the level of noise during the two randomly selected nodes. Then, the energy difference
restructuringprocessofthenetwork. Forclaritywenotethat ∆E = E E between the original g and the new g
ab b a a b
−
our studies concern a class of phenomena that are clearly configurationsiscalculatedandthereallocationiscarriedout
differentfrom the phase transitions investigatedby applying followingtheMetropolisalgorithm[21]. Iftheenergyofthe
models of statistical mechanics originally defined on regu- new graph is lower than that of the original one, the reallo-
lar lattices to an underlying (static) random network struc- cation is accepted; if the new energy is higher, the realloca-
ture[13,14,15],orthephasetransitionsobservedingrowing tionisacceptedonlywithprobabilitye−∆Eab/T. Thisway,in
networks [7, 16], or quasi-static networks [17]. Topological theT limitthedynamicsconvergestoatotallyrandom
→∞
phasetransitionsareaccompaniedbysingularitiesinthether- rewiringprocess,andthus,theclassicalERrandomgraphsare
modynamic functions derived from the partition function of recovered.Ontheotherhand,atlowtemperaturesthetopolo-
the statistical graph ensemble and can be characterized by a gieswithlowestenergyoccurwithenhancedprobability.The
drasticchangeinan appropriateorderparameter. Ourstatis- resultingdynamics,byconstruction,satisfiesthedetailedbal-
ticalensemble(similarlytotheonepresentedinRef.[11]),is ancecondition[21].
definedbyintroducinganenergythataccountsfortheadvan- This network rearrangement is formally equivalent to a
tageorlossduringtherearrangement.Inourdescription,this Kawasaki type lattice gas dynamics with conserved number
energymaydependoneitherglobalpropertiesofthenetwork, ofparticlesmovingonaspeciallattice,whichistheedge-dual
orsinglevertexdegreesaswell. graphofthefullyconnectednetwork[6,22]. Thesitesofthis
TheuseofHamiltonianformalismalsoprovidesageneral latticearethepossibleN(N 1)/2connectionsbetweenthe
−
3
1,2
Φ = Φ = s /M,thenumberofedgesofthelargestcon-
s max
nectedcomponentofthegraphs normalizedbythe total
1 2 max
1,3 numberofedgesM,orΦ = Φ = k /M,thehighestde-
k max
2,3
1,4 greeinthegraphkmax dividedbyM. We alsointroducethe
2,4
correspondingconditionalfreeenergyF(Φ,T)via
4 3
3,4 e−F(Φ,T)/T =Z(Φ,T)= e−Ea/T, (6)
{Xga}Φ
1,5
where g isasubsetof g ,consistingofallthegraphs
1 { a}Φ { a}
with order parameter Φ. A phase transition, where a rapid
change occurs in the order parameter from Φ = 0 towards
5 2 4,5 highervalues,isalsoaccompaniedbyashiftoftheminimum
3,5
1,4 2,5 ofthe conditionalfree energyF(Φ,T). A suddenchangein
3,4
the position of the global minimum signals a discontinuous
2,4
4 3 1,3 2,3 (firstorder)phasetransition,whereasagradualshiftindicates
eitheracrossoveroracontinuousphasetransition.
1,2
Inthenextsectionwebrieflydiscussthenumericalmethods
FIG. 1: Two simple examples of graphs (left hand side) and the usedtostudytopologicalphasetransitions.
corresponding edge-dual graphs(righthandside). Thefullspheres
in the edge-dual graphs represent occupied sites, corresponding to
existingbondsintheoriginalgraph(drawnwithsolidlines),andthe III.NUMERICALMETHODS
hollow spheres are the empty sites, corresponding to absent bonds
(representedbydashedlines)intheoriginalnetwork. Therewiring
III.A)Exactenumerationmethod
ofanedgeintheoriginalgraphsisequivalenttothedisplacementof
thecorrespondingparticleontheedge-dualgraph.
Thenumericalresultsshowninthispaperwereobtainedby
two alternativemethods. Motivatedby the successof a sim-
vertices, and the particles wandering on the sites are the M
ilar approach used in random-walks, percolation, and poly-
edges,asshowninFig. 1.
mersrelatedproblems[23,24,25,26],forsmallsystemswe
Thepartitionfunction(1)containsmanytermscorrespond-
evaluatedthe partitionfunctiontogetherwith the probability
ing to topologically equivalent graphs: these graphs can be
of every individual state via an exact enumeration method.
simply transformedinto one another by an adequate permu-
In this approach, first all possible connected configurations
tation of the indexing. Since we consider energies E that
α with a given number of edges are generated successively up
dependonlyonthetopologyt ,itisnaturaltorewritethepar-
α to M: the graphs with m + 1 edges are constructed from
titionfunctioninaformwherethesummationrunsthroughall
the graphs with m edges either by linking a new vertex to
possibletopologies:
one of the old vertices, or by linking two previously uncon-
nected vertices. Next, all possible configurationscontaining
Z(T)= e−Eα/T. (2)
Nα M edgesare obtainedfrom the combinationof smaller con-
{Xtα} nectedgraphs,withsizesuptoM. Finally, iscalculated
α
N
foreachtopologyt bycountingthenumberofpossibleper-
Hereweintroduced tocountthenumberofconfigurations α
α
N mutationschosentolabeltheverticesinthestate. (Formore
belongingtotopologyt . Expression(2)canberewrittenas
α
detailsandasimpleexample,seeappendixA).Theprobabil-
Z(T)= e−Eα/T+ln(Nα) = e−Fα/T, (3) itypα,ofatopologytαthencanbeobtainedfrom
{Xtα} {Xtα} e−Eα/T
Fα=Eα−TSα, (4) pα = Nα Z . (7)
S =ln( ), (5)
α Nα Once the set of possible states with appropriateprobabilities
hasbeenconstructed,onecanevaluatethe expectationvalue
whereF isthefreeenergyandS istheentropyofthetopol-
α α
ofany thermodynamicquantityusing
ogytα. Q
Weareinterestedinthepossiblesingularitiesinthethermo-
= (t )p . (8)
dynamicfunctionsderivedfromthepartitionfunctionabove, hQi Q α α
since they, if there are any, correspond to phase transitions Xtα
inthetopologyoftheassociatednetworks. Thesetransitions The advantage of this technique, beside producing exact
canbebestmonitoredbyintroducingasuitableorderparam- results, is that the set of has to be calculated only once,
a
N
eter. Asweareprimarilyinterestedinthetransitionsbetween independentlyoftheenergyfunctionsconsidered,incontrast
dispersed and compact states, a natural choice can be either to Monte-Carlo simulations, where the simulation has to be
4
restarted from the beginning every time we introduce a new inthegraph,theothergroupcontainsthesingle-vertexener-
typeofenergy. Thismethodislimitedbytherapidgrowthof gies.
thenumberoftopologieswithM. Fornetworksofsizeseen
intherealworld(M > 102)therealizationofthismethodis
clearlyunfeasible. IV.CLUSTERENERGIES
III.B)Monte-Carlosimulations As mentioned in the introduction, the classical random
graphmodel(correspondingtoT )exhibitsaphasetran-
→∞
Thelatticegasmodeldefinedontheedge-dualgraphofthe sitionwhentheaveragedegreeofthevertices, k =2M/N,
h i
fully connectednetworkrelaxes slowly, because interactions isvariedaround k = 1. For k < 1, thenetworkconsists
h i h i
aredense[31]andtheenergyminimaaresometimeslocalized ofsmall,disconnectedclusters,ontheotherhand,for k 1
h i≥
inhardlyaccessiblepartsofthephasespaceofthesystem. A agiantconnectedcomponentemergesinthegraphcollecting
goodexampleisthetransitionfromaclassicalrandomgraph afiniteportionoftheedges. Nearthecriticalpointthesizeof
–stableathightemperatures–toastar,whichisstableatlow thegiantcomponentscalesas( k 1)M.
h i−
temperatures. ThesimplestMonte-Carlorewiringsimulation Basedonthelatticegasanalogyweexpectthatif k < 1,
h i
(discussedinSec. II.)triestomovearandomlychosenedge thenforasuitablechoiceoftheenergy(onethatrewardsclus-
toarandomlychosennewlocation. However,thismethodis tering)asimilardispersed-compactphasetransitionoccursat
veryinefficient,ifonewouldliketosimulatethecondensation a finite temperature T( k ). Such a transition can be best
h i
ofedgesintoastarinalargesystem. monitoredbytheorderparameterΦs =smax/M (thenumber
Thereare severalsimulationtoolsthatcanhelptoachieve ofedgesinthelargestconnectedcomponentsmax dividedby
fasterconvergence. We haveusedtheso-calledparalleltem- thetotalnumberofedges),oftenusedingraphtheory[28].
pering(alsocalledexchangeMonte-Carlo)methodinseveral Themostobviousenergysatisfyingtheaboverequirement
cases [27]. Except for first-order transitions, this algorithm isamonotonicallydecreasingfunctionE = f(smax). Inthis
can be used well to measure the transition at an acceptable casetheenergyisindependentofthedistributionofthesizeof
speedandhighprecision. smallerclusters,orofthestructuraldetailsofthelargestclus-
The algorithm can be viewed as an improved version of ter: onlythe size ofthe largestclustermatters. Theentropic
simulatedannealing. Severalreplicasofthesystemaresimu- partoftheconditionalfreeenergyinthiscasecanbeestimated
latedsimultaneously,andeachofthemisconnectedtoasep- bycountingthenumberofconfigurationsatgivensmax. The
arate heatbath. A replica in a hotheatbath will explorethe numberofdifferentconnectedconfigurationsofsizesmaxcan
”large-scale” structure of phase space, and the motion of a be estimated as ssmmaaxx to leading order [29], (for an intuitive
replicainacoldheatbathwillberestrictedtoasmallpartof derivationseeappendixB).Thistermhastobemultipliedby
phasespace,whereitwillexplorethedeepbutnarrowenergy thenumberofpossibleselectionsofthesesmaxverticesoutof
wells. N,whichissimplyN oversmax.TheM smaxleft-outedges
−
In the exchange Monte-Carlo method, after a given num- canbeplacedanywherebetweentheN smaxremainingver-
−
berofconventionalupdatestepswithineachreplica,exchange tices,withtherestrictionthattheycannotformclusterslarger
stepsaremade.Tworeplicas(withneighboringtemperatures) thansmax. Since we considersmax = ΦsM to be an exten-
are chosen at random, and a Monte-Carlo-type decision is sivequantityandthetypicalsizeofthelargestcomponentof
madewhetherthetworeplicasshouldbeexchanged,i.e.,their theseleft-outedgesscalesslowerthanM,thisconstraintcan
temperaturesshouldbeswapped. WithMetropolisdynamics, beneglected. Hence,thecontributionfromtheleft-outedges
iftheproductoftheenergydifferencebetweenthetworepli- canbewellestimatedbyan(N smax)2/2over(M smax)
− −
cas(∆E) andthedifferenceofinversetemperatures(∆β) is factor. Ifwecombinethesefactorstogether,theninthether-
positive, then this exchange is accepted, otherwise it is ac- modynamiclimit (when N,M , k = const.) we can
→ ∞ h i
cepted only with probabilitye∆β∆E. Thatis, a replica with write
a high energy and a low inverse temperature (i.e., high tem- N (N Φ M)2/2
(Φ M)ΦsM − s . (9)
perature)will be more likely to remain in its own heat bath, Nclus ≈ s Φ M M Φ M
whereasareplicawithahighenergyandahighinversetem- (cid:18) s (cid:19)(cid:18) − s (cid:19)
Since the energyof the system is a functionof the orderpa-
perature(i.e.,lowtemperature)willbelikelytobe”put”into
rameterΦ itself,theconditionalfreeenergyF(Φ ,T)canbe
aheatbathwithahighertemperature. s s
expressedas
Weshallnowmoveontoreviewsomeofenergyfunctions,
thatleadtophasetransitions,whenthetemperatureischanged e−F(Φs,T)/T = e−f(Φs)/T =e−[f(Φs)−TlnNclus]/T(1.0)
clus
N
from zero to infinity at constant k . Since at T = the
h i ∞ ByusingStirling’sformula[l! (l/e)l√2πl]toapproximate
entropicallyfavorable graphsdominate, a minimum require- ≈
the factorials and neglecting terms of (lnN), for lnN
ment for the energy function is that the configurations with O clus
weget
thelowestenergymustalsohavelowentropy. Wedividethe
investigatedenergyfunctionsintotwocategories. Inthefirst 2M 2M
ln const.+ 1+ln Φ M
categoryweputtheenergiesthatdependoncomponentsizes Nclus≈ N − N s
(cid:20) (cid:21)
5
3M 1 M2 1
+ Φ2M. (11) 3
2N − 2 − N2 s T Φ 1
(cid:20) (cid:21)
By replacing 2M/N with k in the expression above, the
h i
resultingconditionalfreeenergycanbeexpressedas
2
0.5
F(Φ ,T) f(Φ M)+MT [ k 1 ln( k )]Φ
s s s
≈ (h i− − h i
0
Φ2 1 0 1 2 3
+ k 2 3 k +2 s . (12)
1 / T
h i − h i 4 )
h i
The simplest choice for an energy function that depends
onlyonthesizeofthelargestcomponentis 0
0 0.2 0.4 0.6 0.8 1
< k >
f(s )= s = Φ M. (13)
max max s
− −
FIG. 2: The phase diagram and the order parameter for the E =
InthiscaseitcanbeclearlyseenfromEq.(12)thataslongas
smaxenergy. Mainpanel: Thewhiteandshadedareascorrespond
−
totheorderedphase (containingagiantcomponent) andthedisor-
1
T >T ( k )= , (14) dered phase, respectively, as given by Eq. (14). Inset: The order
c
h i hki−1−ln(hki) parameterΦ=Φs =smax/M obtainedfromMonte-Carlosimula-
tionsasafunctionoftheinversetemperaturefor k = 0.1(trian-
thefreeenergyhasaminimumatΦs = Φ∗s(T) = 0,i.e.,the gles)and k =0.5(circles).Eachdatapointisanhenisembleaverage
h i
configurationis dispersed (see main panel of Fig. 2). When of 10 runs, time averaged between t = 100N and 500N Monte-
the temperature drops below T ( k ), the minimum moves Carlosteps. TheopenandclosedsymbolsrepresentN = 500and
c
awayfromΦ = 0andagiantcohmpionentappears. Nearthe 1,000 vertices, respectively. The critical exponent, in agreement
s
withtheanalyticalapproximations(solidlines),wasfoundtobe1.
criticaltemperatureT ( k ),theorderparameteratthemini-
c
h i
mumofthefreeenergycanbeestimatedfromEq.(12)as
appearingbetweencoolingandheatingsupportsthetheoreti-
T−1 T−1( k ) calconsiderationsaboutthecoexistingminimathatindicatea
Φ∗(T)=2 − c h i , (15)
s k 2 3 k +2 firstorderphasetransition,summarizedintheinsetofFig.3.
h i − h i Theanalyticalconditionalfreeenergygainedbysubstituting
indicating that we are dealing with a continuoustopological f(s ) = s2 into (12) has a single minimum, when
max − max
phasetransition(seeinsetofFig.2). T < 80and whenT > 430, in formercase at the dispersed
Topologicalphasetransitionsoffirstorderarealsoexpected state(Φ = 0),inlattercaseattheconnectedstate(Φ = 1).
s s
to occur for other forms of cluster energies. For example, Atintermediatetemperaturesthese two minimacoexist, pre-
whenf(s )startswithazero(orpositive)slope. Insucha dicting a first order phase transition somewhere in the mid-
max
case,thebehavioroftheconditionalfreeenergyatveryhigh dle part of this temperature interval. The transition regime
and very low temperatures is similar to the previous case: 170 < T < 270 observed in the MC simulation is compat-
when T , the energy term f(s ) can be neglected ible with the analytical result, since one does not expect the
max
→ ∞
in (12), and the entropic term has a minimum at Φ = 0, simulationtorevealthemeta-stablestatebesideitsdominant
s
whichis the dispersedstate. In contrast, when T 0, only stablecounterpartin case ofa verysignificantdifferencebe-
→
the energy term remains, resulting a minimum in F(Φ ,T) tweenthedepthsoftheaccordingminimainthefreeenergy.
s
at Φ = 1, the compactstate. However, there is also an in- We have also investigated the case of f(s ) =
s max
termediatetemperaturerange,whereF(Φ ,T)givenby(12) s ln(s ),bothanalyticallyandnumerically.Similarly
s max max
−
starts as an increasing function at Φ = 0 (since the linear tothepreviouscase, itcanbeshownthatthereisatempera-
s
termintheentropydominatesforsmallΦ ),thenreachesits tureregime,wheretheconditionalfreeenergyasafunctionof
s
maximumsomewhereinthe[0,1]intervalandcontinuesasa Φ hastwocompetingminima,hencethetransitionisoffirst
s
decayingfunction from that pointon (since the higher order order.
decaying terms in the energyovercomethe increasing terms SincetheenergyE = f(s )dependsonaglobalquan-
max
atlargerΦ ). As a consequence,the conditionalfreeenergy tity(thesizeofthelargestconnectedcomponent)itmightbe
s
hastwocompetingminimainthe[0,1]interval,ameta-stable also reasonable to define the energy of the graph as E =
and a globally stable. The coexistence of stable and meta- f(s ), where the summation goes over each component
j j
stable minimaat the transitionbetween the phasesis a char- ands denotesthenumberofedgesinthejthone. Thetotal
j
P
acteristic of first order phase transitions. A simple function numberofedges,M = s ,isconservedbythedynamics,
j j
of thistypeis f(s ) = s2 . Forthis choiceofthe en- hencef(s )mustdecreasefasterthanlineartopromotecom-
max − max j P
ergy,ournumericalresultsareshowninFig.3. Thehysteresis pactification. Whenasinglegiantcomponent(containingthe
6
Φs = M1 / M Φs = M1 / M
1 1
F( Φ ,T)
s
0.5 0.5
4e3
0
−4e3
T
−8e3 400
300
−12e3 200
100
0.5 Φs 1
0 0
160 180 200 220 240 120 160 200 240
T T
pFaIGra.m3e:terI,fΦthse=enMerg1y/Mof,thsheogwrsapahfiirsstEor=der−trsa2mnasxit,iotnh.en(Mth1eiosrdtheer F−IG.Ni4=c:1s2iTehneerogryd.e(rMp1ariasmtheetenr,umΦbser=ofeMdg1e/sMin,thfeorlartgheestEcom=-
numberofedgesinthelargestcomponentofthegraph.) Eachpoint ponentandNcisthenumberofcomponentsinagraph.) Eachpoint
givesthevalueofΦs averagedbetweent = 490N andt = 500N shoPwsthevalueofΦsaftert=200N Monte-Carlostepsinagraph
Monte-Carlostepsinagraphstartedatt = 0fromanErdo˝s-Re´nyi started at t = 0 from an Erdo˝s-Re´nyi random graph (o) or a star
randomgraph(o)orastar( ). ThesimulatedgraphhadN = 500 ( ). The simulated graph had N = 500 vertices and M = 125
verticesandM = 125edge×s. Theinsetshowsthebehaviorofthe e×dges. Observethatforintermediatetemperaturestherearetwodis-
analyticalfreeenergyobtainedfrom(12)usingthesameparameters. tant groups of states with high stability (at Φs 0.6 0.9 and
≈ −
Thetemperatureintervalinwhichthetwominima(atΦs = 0,cor- Φs 0 0.15), and Φs values in the region between these two
≈ −
respondingtotheconnectedstateandatΦs = 1, correspondingto rarelyoccur,i.e.,afirstordertransitionwasfound.
the dispersed state) coexist is fully compatible with the numerical
findingsforthetransitionregime.
p(smax) T=7
majorityoftheedges)emerges,itsenergyf(s )dominates 8
theenergyoftheentiregraph,and,asagoodmapaxproximation, p(T, smax) 0.04 9
theaboveanalysisforE = f(smax)canberepeated,leading 0.2
tofirstorderandcontinuousphasetransitions. Forthecaseof 0.02
E = Nc s2,ournumericalresultsarepresentedinFig.4, 0.15
showi−ngaifi=r1stiorderphasetransition. 0.1 0
0 50 100
For thPe E = − Ni=1siln(si) energy – similarly to the 0.05 smax
case of the E = s ln(s ) energy– we founda first-
max max
−P
order transition between the ordered phase (present at low 120
90
tReemsupletrsaatureressh)oawnnditnheFidgi.so5r.dered phase (high temperatures). smax 60 30 0 2 4 6 8T 10 12
V.SINGLEVERTEXENERGYFUNCTIONS
FIG. 5: Distribution of the size of the largest graph component,
N
Nextweturntoanotherimportantclassoftheenergyfunc- lsomwaxt,emifpthereaEtur=es−thediis=tr1ibsuitlino(nsio)fetnheerglayrgisesutsceodm. Mpoaniennptahnaesl.oAnet
tions, where the energies are assigned to the vertices rather maximumatlargevaPluesofsmax,thisistheorderedphase. Athigh
thantotheconnectedcomponentsofthegraph: temperatures,thelargestcomponentissmall:thegraphisdisordered.
Inset. Usingahigherresolution,onecanobservethatthetransition
N fromthelow-temperaturepeak(T =7)tothehigh-temperaturepeak
E = f(k ), (16) (T = 9) happens viaa bimodal distribution(T = 8, indicated by
i
asolid line). This indicates that theconditional free energy of the
i=1
X systemhastwocompetingminimaattheintermediatetemperature,
wherek denotesthedegree(numberofneighbors)ofvertexi. andthetransitionisoffirstorder.Thegraphsusedforthesimulations
i
Thisenergyisconsistentwithadynamicsinwhichthechange hadN =500verticesandM =125edges.Averagesweretakenfor
ofthedegreeofavertexdependsonlyonthestructureofthe 10 (mainpanel) or 200 (inset) simulationruns between simulation
timesof t = 200N and t = 400N Monte-Carlo stepsusing time
graph in its vicinity. The fitness of an individual vertex de-
stepsoft=N.
pendsonitsconnectivity. Themostsuitableorderparameter
forthisclassofgraphenergyisΦ = Φ = k /M. Again,
k max
7
duetotheconservationofthenumberofedges,M = k , spin-likevariablesconnectedton asn =(1+s )/2. The
i i α α α
the single vertex energy f(k ) should decrease faster than energywiththehelpofthespinsisexpressedas
i
P
k ,ifaggregationistobefavored.
−Wieintroduceanalternativeformforsinglevertexenergies: 1 1N(N−1)/2
E= s s s
α β α
−4 − 2
<α,β> α=1
N X X
1
E = g(ki′), (17) N(N 1)(N 2), (21)
−8 − −
i=1 i′
XX
sincethetotalnumberoflatticesitesequalsN(N 1)/2,and
wherei′runsoverallverticesthatareneighborsofvertexi.In thenumberofadjacentpairsoflatticesitesisN(N− 1)(N
thisinterpretation,thefitnessofanindividualvertexdepends 2)/2. ThisissimilartoaferromagneticIsing-mode−linanex−-
ontheconnectivitiesofitsneighbors,andvertexicollectsan ternal magnetic field. If the number of occupiedsites in the
energy g(ki′) from each of its neighbors. These neighbors latticegaspictureequalsthenumberofunoccupiedsites, the
inturnwillallcollectg(ki)fromvertexi,thereforethetotal contributionfromthe externalmagneticfield vanishesin the
contributiontotheenergyfromvertexiiskig(ki). Thus,by Ising-model picture. However, in the thermodynamic limit,
using whereN ,M , k = 2M/N =const.,thiscondi-
→∞ →∞ h i
tioncannotbefulfilled,sincethetotalnumberofsitesscales
f(ki)=kig(ki), (18) asN2,whereasthenumberofparticlesscalesasM withthe
systemsize.
thetwoalternativeformsofthesinglevertexenergy,(16)and Fortheparticularformoff(k )chosen,thetopologywith
i
(17),becomeequivalent. the lowest overallenergyis a “star” (for simplicity, we con-
siderM <N),wherealltheM edgesareconnectedtosingle
V.A)TheenergyE = k2: mappingtotheIsing-model node.Theformoftheconditionalfreeenergyinthiscasecan
− i
beestimatedasfollows. IntheΦ > 1/2regime,wherethe
P k
Anaturalchoicefortheenergyofsinglevertextypeisthe systemcontainsastarofsizelargerthanM/2,theenergyof
following.Assignthenegativeenergy Jtoallpairsofedges this star dominates the rest of the graph. Therefore, in the
−
thatshareacommonvertexatoneend. Thetotalenergyofa thermodynamiclimit,weneglectthislattercontributiontothe
givenconfigurationisthen totalenergyandapproximatetheenergyofthegraphbythat
of the largest star. Now we estimate the number of possible
E = J N k (k 1)= J N k2+ 1JM, (19) configurationsforagivenvalueofΦk. Incaseofastarwith
−2 i i− −2 i 2 K = ΦkM arms, the central vertex can be chosen from N
Xi=1 Xi=1 differentvertices. Oncethisisfixed,K edgeshavetobedis-
tributed among the N 1 possible links between the other
corresponding to f(ki) = −(J/2)ki2, (or equivalently, to verticesand thecentral−one, yieldinga factorof N 1 over
g(ki) = −(J/2)ki). Theconstanttermin(19)doesnotplay K. The rest of the edges that are not part of the sta−r can be
anyroleinthedynamics,henceitcanbeomitted. Thisform
placed anywhere between the N 1 (non-central) vertices,
oftheenergyisinfullanalogywiththeusualdefinitionofthe −
contributingafactorof(N 1)(N 2)/2over(M K).
energy − − −
N 1 (N 1)(N 2)/2
(K) N − − −
E = J n n (20) Nstar ≈ K M K ≈
− α β (cid:18) (cid:19)(cid:18) − (cid:19)
<Xα,β> N N2/2
N . (22)
K M K
ofalatticegasontheedge-dualgraphofthefullyconnected (cid:18) (cid:19)(cid:18) − (cid:19)
network with nearest neighbor attraction. The summation Again,weuseStirling’sformulatoapproximatethefactorials,
hererunsoveralladjacentpairsoflatticesites(corresponding andneglecttermsoftheorder (lnN)yielding
O
topossibleedgesbetweentheverticesoftheoriginalgraph),
N N N2 N2
and nα = 1 if site α is occupied and 0 otherwise. When ln star M ln + ln
thisenergyisappliedtolattices, werecoverthestandardlat- N ≈ "M M 2M 2M
tice gasmodelof nucleationof vapors. The negativeenergy N N
Φ ln Φ
unit J associatedwithapairofedgessharingavertexinthe k k
− M − M −
origi−nalgraphisequivalenttothebindingenergybetweenthe (cid:18) (cid:19) (cid:18) (cid:19)
Φ lnΦ (1 Φ )ln(1 Φ )
corresponding occupied nearest neighbor sites on the edge- − k k− − k − k
dual graph. By measuring the energies(and temperature)in N2 N2
1+Φ ln 1+Φ (.23)
unitsofJ wecansetJ =1,withoutloosinggenerality.Thus, −(cid:18)2M − k(cid:19) (cid:18)2M − k(cid:19)#
fromnowonJ willbeomitted.
Inthethermodynamiclimit,toleadingorderwereceive
Thelatticegasrepresentationcanbefurthertransformedto
anIsing-model-representationbyintroducingthes [ 1,1] ln (x) Φ Mln(N), (24)
α star k
∈ − N ≈−
8
Φ −20
1 T=0.4
exact
theory
0.6 −40
0.2
−60
100 −80
N
10000 −102
1 0.001 T=1.7 exact
1000 F(Φ ,T)
T k theory
−110
FIG.6: TheorderparameterΦ = Φk = kmax/M asafunctionof
thetemperatureandthesystemsizeforE = i−ki2/2andhki=
0.5. Thesimulationswerestartedeitherfromastar(corresponding
toT = 0,solidline)oraclassicalrandomgrPaph(T = ,dashed −118
∞
line).Eachdatapointrepresentsasinglerun,timeaveragedbetween
t=100N and200N Monte-Carlosteps.Thethicksolidlineshows −150
T=3.0
theanalyticallycalculatedspinodalT1 =M/ln(N).
wheretheΦk independenttermsweredropped.Theresulting −170
conditionalfreeenergyisexpressedas:
F(Φk,T) f(ΦkM)+ΦkMTln(N). (25) exact
≈ −190
theory
Inthepresentcase,withthef(k )= k2energy,(25)canbe
i − i
writtenas
0 0.2 0.4 0.6 0.8 1
Φ
F(Φ ,T) M Φ2M +Φ T ln(N) . (26) k
k ≈ − k k
Note that this approximat(cid:2)ionwould be valid eve(cid:3)n for Φ < FIG.7: Thepictureoftheconditionalfreeenergyatthreedifferent
1/2, if the energy of the graph was simply defined as Ek = temperaturesforthef(ki) = −ki2 energy, obtainedfromtheexact
enumerationmethodplottedtogetherwiththepredictionofoursim-
f(k ).
max pletheoreticalanalysisforM = 12,N = 48. Atlowtemperatures
The parabola given by Eq. (26) has a maximum at Φk = F(Φk,T)isadescendingfunctiononthe[0,1]withaminimumat
T/Mln(N). When T 0, this maximum also shifts to- Φk = 1, the star configuration (top figure), on the other hand for
→
wardszeroandF(Φ ,T)becomesadescendingparabolaon hightemperatures, itbecomesascendingformostpart,withamin-
k
the [0,1] interval. This means that the minimum of the free imumatlowΦk,thedispersedstates(bottompicture). Thereisan
intermediatetemperatureregimeinbetween,wherethemaximumof
energy is at Φ = 1, the star configuration. In contrast,
when the tempekraturegoesabovethe T = M/ln(N) spin- F(Φk,T) separates two competing minima (middle figure), hence
1 thisphasetransitionisoffirstorder.
odalpoint(thicksolidlineinFig.6),themaximummovesout
of the [0,1]intervaland the freeenergybecomesan ascend-
ing parabola, resulting in a minimum at a low value of Φ was evaluated via the exact enumeration method for various
k
(correspondingtoanERrandomgraph). However,thisvalue temperatures, and was found to be in qualitative agreement
cannotbededucedfromEq.(26),becauseitisavalidapprox- with the prediction of the theoretical analysis, as demon-
imationonlyforΦ >1/2.Forintermediatetemperaturesthe strated in Fig. 7. The three different temperature regimes
k
maximumof the parabola separates the two extremetopolo- describedinthepreviousparagraphcanberecognizedinthe
gies: thedispersedrandomgraphandthestar). Oneofthese behaviorof the exactF(Φk,T)as well. Furthermore,in the
two extremestatesismeta-stableandthe otheroneisstable. intermediate temperature regime, where the conditional free
Due to the limited validity of Eq. (26), the stability of these energy has two competing minima, the spinodal curve can
configurationscanbestudiedonlyfortemperatureswherethe also be constructed as is shown in Fig. 8. For large enough
maximumoftheparabolaiswellinsidethe[1/2,1]interval. systems, in MC simulations a sudden change of the order
The scenario of the transition from a dispersed state to parameterbetweenzeroandonecanbeobservedasshownin
the star configuration (see above) indicates that it is a first Fig. 6. Thehysteresisappearingbetweencoolingandheating
orderphase transition. This is well supportedby the results isconsistentwithafirstordertransition.
of both the exact enumeration method and Monte-Carlo
simulations. For small systems, the conditional free energy V. B) The energy E = k lnk : continuous phase-
i i
−
P
9
1 a) 1000
1−P(k)
0.8 min. of F(Φ k ,T) kmax 1 c)
0.6 max. of F(Φ k ,T) 0.01
Φ
100
k
0.4
1 10 100
k
0.2
min. of F(Φ k ,T)
10
0
1 1.2 1.4 1.6 1.8 2
T 0.4 0.8 1.2 1.6
b) T
FIG. 8: The spinodal curve obtained from the exact enumeration
method with E = − iki2, for M = 12,N = 48. At low and
hightemperatures, theconditional freeenergyF(Φk,T)hasasin-
gleminimum (plottedPwithsquares). At intermediatetemperatures
(inbetweenthetwodottedlines)therearetwocompetingminima.
Inthislattertemperatureregime, thespinodal curve isobtained by
plotting the maximum of F(Φk,T)(represented by stars), besides
thetwominima. FIG. 9: Phases of the graph when the energy is E =
− ikiln(ki). (a)Thelargestdegreekmax forN = 10,224ver-
ticesandM =2,556edges.Eachdatapointrepresentsasinglerun,
transition timPeaveragedbetweent = 5,000N and20,000N MCsteps. The
datapointsareconnectedtoguidetheeye.Thereisasharp,continu-
Another application-motivated choice for the single ver- oustransitionnearT =0.85andafirst-ordertransition(withahys-
tex energy is f(k ) = k ln(k ), or equivalently, g(k ) = teresis)aroundT =0.5 0.6.(b)Thethreedifferentplateausin(a)
ln(k ), inspiredi, in pa−rt,iby thie logarithmic law of seinsa- correspondtodistincttop−ologicalphases: kmax = (1)totheclas-
i O
t−ion. It is the logarithm of the degree of a vertex that its sicalrandomgraph,kmax =O(M)tothestarphase(asmallnumber
neighbors can sense and benefit from. In this case the con- ofstarssharingmostoftheirneighbors)andkmax =O(√M)tothe
fullyconnected subgraph. (c)The(cumulative) degreedistribution
figuration with the lowest energy is a fully connected sub-
atT =0.84andt=600N followsapowerlaw.Thisshowsthatthe
graph [or almost fully connected if M cannot be expressed degreedistributiondecaysasapower-lawwiththeexponentγ 3.
as n(n 1)/2]. On the other hand, the star configuration ≈
−
isalso quitefavorable, sincethe energyof boththe maximal
indicatingthatitisacontinuousphasetransition.
possiblestarandofthemaximalpossiblefullyconnectedsub-
ForΦ >1/2Eq.(25)canbeusedagainasagoodapprox-
k
graphscalesas MlnM toleadingorder. Amongstthesub-
− imation for the free energy of the graph, since the compact
dominanttermsintheenergy,thereisadifferenceintheorder
clusterarisingfromthedispersedstate isratherstarlike. By
of √Mln√M between the two, in favor of the fully con-
plugging f(Φ M) = (Φ M)ln(Φ M) into that expres-
k k k
nected subgraph. As before, we choose the order parameter −
sion,weget
to be Φ = Φ = k /M, since this can easily distinguish
k max
bcoetnwneeecntedthseusbegtrwaophcoconufingtuinragtiMonse:dkgmesa,xa≈nd√k2M forMa ffourllya F(Φk,T)≈M(T −1)ln(N)Φk (27)
max
≈
star. toleadingorder,whichislinearinΦ . Inagreementwithour
k
Our MC simulations demonstrate (Fig. 9) that as we cool observationsabove, this formulapredictsthat forT < 1 the
downthesystem,firsttheedgesofthedispersedrandomgraph starisastableconfiguration(Φ =1isaminimumofthefree
k
assembletoformaconfigurationwithafewlargestars(shar- energy),andforT >1itbecomesunstable. Thetransitionat
ing mostof their neighbors),and then at lower temperatures T = T = 1 is thus step-like with no hysteresis, indicating
c
the graph is rearranged into an almost fully connected sub- a continuousphase transition with an infinitely large critical
graph. This is consistent with the fact that beside the slight exponent. WeassumethattheobserveddeviationofT from
c
energeticaldisadvantage,thestarconfigurationisentropically 1intheMCsimulationsisafinitesizeeffect.
more favorable when compared to the fully connected sub-
graph;thereforethelatterconfigurationcantakeoveronlyat V.C)Relationtogrowthwithpreferentialattachment
verylowtemperatures. Thehysteresisnearthefewlargestar
vs. fully connected subgraph transition suggests that it is a AremarkablefeatureoftheMCdynamicsisthatincaseof
firstorderphasetransition. Ontheotherhand,thetransition the energy f(k ) = k lnk , by crossing T from above,
i i i c
−
betweenthedispersedstateandthefewlargestarsisaccom- a scale-free graph (with a degree distribution k−γ with
∼
panied by a singularity in the heat capacity (also seen with γ 3) appears at some point of the evolution of the graph
≃
theexactenumerationmethod),andnohysteresisisobserved, fromtherandomconfigurationtowardsthestar. Thissupports
10
the notion that scale-free graphs are temporary (dynamical) VI.THEGRANDCANONICALENSEMBLE
configurations, not typical in equilibrium distributions. The
MCdynamicsisgovernedbythechangeofthe energyasso- Inthegrandcanonicalensemble,thedegreedistributioncan
ciatedwiththereallocationofanedge.Estimatingtheenergy beexpressedas[11]
changeofavertexbythederivativeofthesinglevertexenergy
f(k ) = k ln(k ),weget∆E = 1 ln(k ). Pluggingthis e−βf(k)−µk
i i i i P =C . (28)
− − k
intotheBoltzmannfactor,exp[ ∆E/T],atT = T = 1we k!
c
−
getaquantityproportionaltoki fortheacceptation/rejection whereCis a normalizationfactorandthe chemicalpotential
ratioof a randomlyselectedmove. Sincethe preferentialat- µ is adjusted to give the correct k . For f(k) = kln(k),
h i −
tachmentin the Baraba´si-Albertmodel[4] is proportionalto usingStirling’sformula,thedistributiontakestheform
k , it is natural that our dynamics also produces scale-free
i
e−(µ−1)k
graphs. P =C k(1/T−1)k. (29)
k
√2πk
Anotherinterestingaspectofthef(k )= k lnk energy When T > 1, this has a tail, which decays faster than ex-
i i i
−
isthattheconfigurationsinthetwocompactphasesresemble ponential,consequently,each vertexhasa small degree. For
the two major graph topologies obtained in Ref. [18], by T < 1,ontheotherhand,thetailbecomesdivergent,signal-
optimizingthenetworkforlocalsearchwithcongestion. Our inga phasetransition atT = T = 1. Notehoweverthat in
c
intermediate phase with a few large central hubs sharing theT <1temperaturerange,duetothenon-extensivecontri-
neighbors is similar to the optimal topology for a small butionofthedivergingdegrees,theensemblesarenotequiv-
number of parallel searches, whereas the low-temperature alent,andthegrandcanonicaldescriptionlosesitsvalidity.
configuration, the fully connected subgraph resembles the Atthecriticaltemperature,thegrandcanonicaldescription
homogeneoustopologyoptimalforalargenumberofparallel might still be valid. Choosing a more general single ver-
searches. However, an important difference between the tex energy, f(k ) = (k α)ln(k ), and setting k such
i i i
− − h i
two problems is that in our case a vertex is allowed to lose thatµ = 1, the degreedistributionacquiresa powerlaw tail
all of its connections under the restructuring process. The (P k−(α+1/2))andthenetworkbecomesscale-freeatthis
k
∼
two “similar-to-optimum”configurationsappear as a natural temperature. Wehavetostressthoughthatthescale-freenet-
consequence of the underlying dynamics. This observation workatT isnotgeneral: forµ > 1thetaildecaysexponen-
c
suggests a potential application of the presented theory: tially,andforµ<1thetaildiverges.
tackling problemsrelated to graph topologyoptimizationby
simulatedannealingtechniques.
VII.SUMMARY
We studied the restructuring in networks using a canoni-
V.D)Topology-dependentnon-extensivenessoftheenergy
cal ensemble, where temperature correspondsto the level of
noise in real systems andthe energyassociated with the dif-
ferentconfigurationsaccountsfortheadvantagegainedorlost
Bothtypesofthesinglevertexenergyfunctionsdiscussed
during the rewiring of the edges. We found that for vari-
in the present section lead to compact configurations at low
ous types of energies, first order and continuous phase tran-
temperatures, for which the most highly connected vertices
sitions may appear when changing temperatures. In case of
possess macroscopic numbers of edges. As a consequence,
the E = s energy,if k < 1, a dispersed-loosephase
max
the energy of the system scales differently with system size − h i
transitionoccursatafinitetemperature,equivalenttotheper-
at high and low temperatures, and diverges differently as
colationphasetransitionofclassicalrandomgraphswhen k
N . Athigh temperature,the system consistsof many h i
is varied around k = 1. We obtained a simple expression
→ ∞
small unlinked clusters of about the same size, therefore a h i
for the T ( k ) critical line separating the two phases in the
c
changeinthetotalsystemsizeaffectsonlythenumberofthe h i
[ k ,T] plane from a theoretical analysis of the conditional
clusters,andtheenergyscalesasN. Ontheotherhand,when h i
freeenergy. For otherformsof the energydependingon the
f(k )= k ln(k ),atlowtemperaturestheenergyofthestar
i i i size of the largest cluster we found first order phase transi-
−
andthefullyconnectedsubgraphscalesasNln(N); incase
tions. We also studied the effects of different single vertex
o(ufnfli(kkei,)i.=e.,−inkit2h,ethmeeeanne-rfigeyldofIstihnegsmtaordsecla),letshearseNis2n.oTwhauys energies,namelytheE = − iki2 andE = − ikiln(ki)
cases. The network in the former case exhibits a first or-
tochooseanappropriatecouplingconstantthatcouldrender P P
derphasetransitionfromadispersedstatetoastar-likestate,
theenergyextensiveinalltopologicalstatessimultaneously.
wherenearlyalledgesarelinkedtoasinglevertex. Withthe
k ln(k ) energy, the dispersed state transforms into a
− i i i
Nevertheless,thedispersedstate(havinganextensivegraph compact one with a few large stars via a continuous phase
P
energy)canequallybestudiedinthegrandcanonicalensem- transition. Inthecriticalpoint,scale-freenetworkscanbere-
ble. covered.Atlowertemperaturesanothertransitionoccurs(this