Singularities in ternary mixtures of k-core percolation

0.34 MB·English

by  Davide Cellai

Singularities in ternary mixtures of k-core
percolation
DavideCellaiandJamesP.Gleeson
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1 AbstractHeterogeneousk-corepercolationisanextensionofapercolationmodel
whichhasinterestingapplicationstotheresilienceofnetworksunderrandomdam-

] age. In this model, the notion of node robustness is local, instead of global as in
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uniformk-corepercolation.Oneoftheadvantagesofk-corepercolationmodelsis
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- the validity of an analytical mathematical framework for a large class of network
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topologies.Westudyternarymixturesofnodetypesinrandomnetworksandshow
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d the presence of a new type of critical phenomenon. This scenario may have use-
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t fulapplicationsinthestabilityoflargescaleinfrastructuresandthedescriptionof
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m glass-formingsystems.
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o 1 Introduction
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Percolation, with its many modifications and extensions, is a problem with a ven-

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erable past and many applications in the most diverse disciplines [18]. With the
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rapiddevelopmentofnetworkscience,severalnewproblemsandmodelshavebeen
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introduced. Among the different directions, k-core percolation constitutes a devel-
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5 opment which, in spite of its somewhat simple definition, is able to encapsulate a
4 numberofinterestingproblemswhichcanoftenbeapproachedwitharobustmath-
1. ematicalformalism.Givenanetwork,ak-coreisdefinedasthesubnetworkwhere
0 eachnodehasatleastk neighboursinthesamesubnetwork.Ak-corecanequiva-
3 lentlybedefinedasthesubnetworkremainingafteracullingprocessconsistingin
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recursivelyremovingallthenodeswithdegreelowerthank.k-corepercolationhas
:
v
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X DavideCellai
MACSI,DepartmentofMathematicsandStatistics,UniversityofLimerick,Ireland,e-mail:da-
r
[email protected]
a
JamesP.Gleeson
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland, e-mail:
[email protected]
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2 DavideCellaiandJamesP.Gleeson
applicationsinmanydifferentdisciplinesincludingjamming[15],neuralnetworks
[8],granulargases[1],evolution[14],socialsciences[13]andthemetal-insulator
transition[4].Theapplicationofk-corepercolationweareinterestedinhere,how-
ever,referstothestabilityofanetworkunderrandomdamage.Afraction(1−p)of
nodesisremoved(togetherwiththeadjacentedges)andthesizeM ofthelargest
k
k-core cluster is studied as a function of p. The k-core strength M can vanish ei-
k
ther continuously or discontinuously, a scenario that can be described in terms of
phase transitions (second and first order, respectively) and can be important in the
stabilityoflargescaleinfrastructures[9,10].Arecentextensionofk-corepercola-
tion,namedheterogeneousk-core(HKC)percolationconsidersthethresholdk asa
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localproperty,andallowsthereforethemixingofdifferentcriticalphenomena.An
analyticalformalismhasbeenintroducedtoapproachthismodelonnetworksand
binary mixtures of thresholds k have been investigated [2, 5, 6]. On a wide class
of networks, this model displays phase diagrams which are topologically equiva-
lent to the ones calculated for a recently introduced spin model of glass-forming
systems [17, 16]. This model can be seen as a heterogeneous development of the
Fredrickson-Andersen (FA) model of facilitated spins, where the facilitation con-
sists of a local constraint in the number of spins down in order for the considered
spintobeabletoflip[11,17].Ithasbeenshownthatthismodelreproduceschar-
acteristic signatures of glass-forming systems called glass-transition singularities,
which correspond to distinctive critical phenomena for the appropriate choice of
theparameters[12,16].Oneoftheglass-transitionsingularitieswhichhasnotyet
been explored is the so-called A singularity, characterised by the coalescence of
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two critical points. As the HKC model appears to reproduce all the relevant criti-
cal phenomena for binary mixtures, it is interesting to look for this singularity as
well.Thepurposeofthispaper,therefore,istoinvestigate,forthefirsttime,ternary
mixturesinHKCpercolationandcalculatethecriticalpointcorrespondingtoanA
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singularity.
In Section 2 we present the analytical formalism used here (from [2]), in Sec-
tion 3 we sketch the behaviour of the model for binary mixtures, in Section 4 we
showacaseofternarymixturewithanA singularityandinSection5wegivethe
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conclusions.
2 Formalism
Letk bethek-corethresholdofthenodei.Afraction(1−p)ofnodesarerandomly
i
removed:theproblemconsistsincalculatingthesizeoftheHKC(ifitexists).We
consider the configuration model of random networks, defined as the maximally
randomnetworkwithagivendegreedistributionP(q).Theconfigurationmodelhas
thepropertyofbeinglocallytree-like,i.e.thenumberoffiniteloopsvanishesfor
infinitenetworks.Thispropertyallowstoconsider,inaninfinitenetwork,theHKC
equivalenttothe(k −1)-arysubtree,definedasthetreeinwhich,aswetraverseit,
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each encountered vertex has at least k −1 child edges. Then we can write a self-
i
Singularitiesinternarymixturesofk-corepercolation 3
consistentequationforZ,theprobabilitythatarandomlychosennodeistherootof
a(k −1)-arysubtree[2]:
i
∞ qP(q) q−1 (cid:18)q−1(cid:19)
Z = pr ∑ ∑ Zl(1−Z)q−1−l+
(cid:104)q(cid:105) l
q=ka l=ka−1
∞ qP(q) q−1 (cid:18)q−1(cid:19)
+ps ∑ ∑ Zl(1−Z)q−1−l+
(cid:104)q(cid:105) l
q=kb l=kb−1
∞ qP(q) q−1 (cid:18)q−1(cid:19)
+p(1−r−s) ∑ ∑ Zl(1−Z)q−1−l. (1)
(cid:104)q(cid:105) l
q=kc l=kc−1
wherethethreethresholdsk=(k ,k ,k )arerandomlyassignedtonodeswithprob-
a b c
abilityr,sand(1−r−s),respectively.Inthispaperwealwaysassumek ≥2,so
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wedonotneedtoconsiderthecasewheretheremaybefiniteclustersintheHKC.
Duetotheabsenceoffiniteloops,nofiniteHKCclusterscanexistifk ≥2forall
i
nodesi[10,2].
We can then write the probability M that a randomly chosen node is in the
abc
HKC,foramixtureofthreetypesofnodes:
∞ q (cid:18)q(cid:19)
M (p)= pr ∑ P(q) ∑ Zl(1−Z)q−l+
abc
l
q=ka l=ka
∞ q (cid:18)q(cid:19)
ps ∑ P(q) ∑ Zl(1−Z)q−l+
l
q=kb l=kb
∞ q (cid:18)q(cid:19)
p(1−r−s) ∑ P(q) ∑ Zl(1−Z)q−l. (2)
l
q=kc l=kc
3 Binarymixturesinheterogeneousk-corepercolation
In homogeneous k-core percolation it is known that, for networks with a fast de-
creasingdegreedistribution,namelyP(q)<1/qγ withγ>3forq→∞,arandomly
damaged network collapses continuously for k≤2 and discontinuously for k≥3
[7, 3, 10]. In recent papers, it has also been shown that in the case of binary mix-
tures HKC percolation is characterized by a few different critical phenomena. For
example, in mixtures where the two values of k are associated to continuous and
discontinuoustransitions,respectively,acriticalpoint(alsocalledA singularityor
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cusp singularity) is usually observed [2, 6]. That is the case, for instance, of the
mixturesk=(1,3)ork=(2,4),wherethelineoffirstordertransitionsendsina
critical point and a line of second order transition intersects the former. The case
k=(2,3), though, is quite peculiar as the critical line exactly matches the line of
firstordertransitionsgivingrisetoatricriticalpoint(Fig.1)[5].
4 DavideCellaiandJamesP.Gleeson
1.0
perc.
0.8
0.6 (2,3)
r
0.4
0.2
(2,4)
(1,3)
no perc.
0.0
0.0 0.1 0.2 0.3 0.4 0.5
p
c
Fig.1 Comparisonofthephasediagramsofthecasesk=(1,3),k=(2,3)andk=(2,4)for
Erdo˝s-Re´nyi graphs with z1 =10. Continuous lines represent second order phase transitions,
whereas dashed lines represent first order phase transitions. The red dot represents a tricritical
point;theotherdotsarecriticalpoints.IntermsofthenotationgiveninSection2,abinarymixture
correspondsforexampletos=1−r.
When both values of k are characterised by a first order transition, instead, the
phasediagramdisplaysacriticalpointonlywhenthetwovaluesofk aredifferent
enough[6].Fig.2,forinstance,showsthatthemixturek=(3,8)hasacriticalpoint
whereask=(3,4)doesnothaveone.
4 TheA singularity
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From the analogy between HKC percolation and facilitated spin models, it is in-
teresting to investigate whether the singularities observed in models of the glass-
formingsystemsarealsopresentinHKCpercolation.Inparticular,thefactthatall
the critical phenomena observed so far in HKC percolation are in the same class
of universality as the ones observed in the FA model with heterogeneous facilita-
tion is quite remarkable [16, 6]. The MCT of glass-forming systems also predicts
an “A singularity”, meaning the coalescence of a critical point into a line of first
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orderphasetransitions[12].TheA singularityisalsonamedaswallow-tailbifur-
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cation.Asthisscenarioisassociatedwithathree-parametertheory,itisreasonable
Singularitiesinternarymixturesofk-corepercolation 5
1.0
0.8
0.6
r
0.4
0.2 (3,4) (3,8)
0.0
0.0 0.1 0.2 0.3 0.4 0.5
p
c
Fig.2 Phasediagramofthecasesk=(3,8)andk=(3,4)fortheErdo˝s-Re´nyigraph(z1=30).
SymbolsareasinFig.1.Whilefork=(3,4)thereisonlyoneline,inthecasek=(3,8)thereare
twolinesofdiscontinuoustransitionsandacriticalpoint.
toexpectthatthecorrespondingsingularityinourHKCmodelcanbeobservedin
athree-componentmixture.Moreover,wehaveseeninSection3thatbinarymix-
tureswithk ≥3areeithercharacterisedbyasinglelineoffirstordertransitionsor
i
twolinesoffirstordertransitionswithasinglecriticalpoint.Itisthereforenatural
toconsideraternarymixturewhichinterpolatesbetweenthetworegimes.Wenow
showthatanA singularityisindeedpresentintheternarymixturek=(3,5,8)on
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Erdo˝s-Re´nyigraphs.
Equation(1)canbere-writtenas
1
f (Z)= , (3)
358 p
where
(cid:40) (cid:34) (cid:35)(cid:41)
1 3 (z Z)n 6 (z Z)n
f (Z)= 1−e−z1Z 1+rz Z+s∑ 1 +(1−r−s)∑ 1 . (4)
358 Z 1 n! n!
n=1 n=1
Ateveryfixedfractionofremainingnodes p,dependingonthevaluesoftheparam-
eters r and s, f has one or two maxima in Z. If the second maximum is higher
358
thanthefirstone,thereisonlyonefirstordertransitionbetweenapercolatingand
a non-percolating HKC. If the first maximum is higher than the second one, there
6 DavideCellaiandJamesP.Gleeson
aretwofirstordertransitions:onebetweenahigh-kandalow-kphaseandanother
towardsacollapsingHKC.Typically,thesecondmaximumdisappearsatacritical
point,whichcanbefoundbyimposingthecondition:
f(cid:48) (Z)= f(cid:48)(cid:48) (Z)=0. (5)
358 358
Thisdefinesalocusinthe(r,s)planecorrespondingtoalineofcriticalpoints.At
high values of s, however, the critical point disappears and only a single line of
first order transitions survives. This is due to the fact that the 5-nodes interpolate
betweentheothertwovaluesofkandthereisnocriticalpointineitherthemixture
k=(3,5)ork=(5,8).TheconditionofanA singularitycorrespondstotheonset
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ofthisbehaviourandisdefinedbythecondition:
f(cid:48) (Z)= f(cid:48)(cid:48) (Z)= f(cid:48)(cid:48)(cid:48) (Z)=0, (6)
358 358 358
whichyields
r =0.3140687639806 s =0.1831697392197 (7)
∗ ∗
which is the position of the A singularity on the phase diagram. Fig. 3 shows the
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phasediagramintheplane(p,r)forafewvaluesofs.
The critical exponent β is defined by the vanishing of the order parameter in
approaching a phase transition: M (p)−M (p ) ∼ (p−p )β. As in binary
358 358 c c
mixtures, the critical exponent at the discontinuous phase transition is β =1/2,
whereasitbecomesβ =1/3atthecriticalpoint.InthevicinityoftheA singular-
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ity,wehave(p−p )∼ f (Z )−f (Z)∼(Z−Z )4,fromwhichitfollowsthat
∗ 358 ∗ 358 ∗
M (p)−M (p )∼(p−p )1/4,asM islinearin pandZinthevicinityofthe
358 358 ∗ ∗ 358
transition.Thus,thevaluesofthecriticalexponentβ canbesummarizedasfollows:

1/2hybridtransitionr(cid:54)=r ,s(cid:54)=s
 c ∗
β = 1/3criticalpoint r=rc,s(cid:54)=s∗ (8)
1/4A singularity r=r ,s=s
4 ∗ ∗
Theuniquevalueofβ=1/4identifiesthecriticalpointasaA singularity.Thisphe-
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nomenonhasaninterestinginterpretation.Inanetworksformedbytypesofnodes
ofverydifferentfragilities(asink =3,k =8),thecollapseduetorandomdamage
a c
canbecatastrophic,withthepresenceofmultiplediscontinuoustransitions.Thein-
troductionofnodesofintermediatefragility(k =5)canweakenthecriticalpointup
b
todisappearance,makingasignificantregionofthephasediagramstable(Fig.3).
Thispurelymathematicalremarkcouldbeusefulinprincipleinplanningthestabil-
ityoflargescaleinfrastructures.Anotherinterestingapplicationisinmodellingthe
glasstransition.ThepresenceofA singularitiesispredictedbythemode-coupling
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theoryofglass-formingsystems[12].DuetothestrongsimilaritiesbetweenHKC
percolationandfacilitatedspinmodels,itisreasonabletoexpectthatthissingular-
ityshouldalsobepresentinsuitableternarymixturesoffacilitationparametersin
suchspinmodels[17,16].
Singularitiesinternarymixturesofk-corepercolation 7
0.8
perc.
0.6
r 0.4
0.2 s = 0.5 s = s s = 0
∗
no perc.
0.0
0.1 0.2 0.3 0.4
p
c
Fig.3 Phasediagramofthemixturek=(3,5,8)forErdo˝s-Re´nyigraphs(z1=30).randsare
thefractionsofnodesoftype3and5,respectively,and pcisthefractionofundamagednodesat
aphasetransition.Linesofdiscontinuoustransitionsareplottedintheplane(p,r)atthreefixed
valuesofs.Intheabsenceoftype5(s=0),therearetwolinesoffirstordertransitions.Oneof
themendsinacriticalpoint(blackdot).Atthecriticalfractions∗,thetwolinestouchataA4
singularity(redstar).Fors>s∗,thereisonlyasinglelineofdiscontinuoustransitions.
5 Conclusions
Inthispaperwehaveappliedtheformalismofheterogeneousk-corepercolationto
ternarymixturesofthresholdsk.Inparticular,wehavecalculatedthephasediagram
ofaternarymixtureforErdo˝s-Re´nyigraphs.Thismixturedisplaysacharacteristic
A singularity,i.e.acriticalphenomenoncharacterisedbythemergingofacritical
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point with a distinct line of discontinuous transitions. A peculiarity of this point
is the change in the critical exponent β of the order parameter, which uniquely
assumesthevalue1/4.Thecharacteristicsofthestudiedphasediagramsmaygive
usefulinformationindesigninglargescaleinfrastructurestoberesilienttorandom
damage.Thissingularityisalsopredictedbymodelsoftheglasstransition[16]and
ourmodelappearstobeinthesameuniversalityclassofkineticspinmodelswith
heterogeneousfacilitation.
Acknowledgements WeacknowledgeusefuldiscussionswithMauroSellitto.Thisworkhasbeen
fundedbyScienceFoundationIreland,grants:11/PI/1026and06/MI/005.
8 DavideCellaiandJamesP.Gleeson
References
1. Alvarez-Hamelin,J.I.,Puglisi,A.:Dynamicalcollisionnetworkingranulargases.Phys.Rev.
E75,51302(2007)
2. Baxter,G.J.,Dorogovtsev,S.N.,Goltsev,A.V.,Mendes,J.F.F.:Heterogeneousk-coreversus
bootstrappercolationoncomplexnetworks. Phys.Rev.E83(5),051134(2011)
3. Branco,N.S.:Probabilisticbootstrappercolation. J.Stat.Phys.70,1035(1993)
4. Cao,L.,Schwarz,J.M.:Quantumk-coreconductiononthebethelattice. Phys.Rev.B82,
104211(Sep2010),http://link.aps.org/doi/10.1103/PhysRevB.82.104211
5. Cellai, D., Lawlor, A., Dawson, K.A., Gleeson, J.P.: Tricritical Point in Hetero-
geneous $k$-Core Percolation. Physical Review Letters 107, 175703 (Oct 2011),
http://dx.doi.org/10.1103/PhysRevLett.107.175703
6. Cellai, D., Lawlor, A., Dawson, K.A., Gleeson, J.P.: Critical phenomena in heterogeneous
k-corepercolation(Sep2012),http://arxiv.org/abs/1209.2928
7. Chalupa,J.,Leath,P.L.,Reich,G.R.:Bootstrappercolationonabethelattice. J.Phys.C12,
L31(1979)
8. Chatterjee,N.,Sinha,S.:Understandingthemindofaworm:hierarchicalnetworkstructure
underlyingnervoussystemfunctioninc.elegans. Prog.BrainRes.168,145–153(2007)
9. Cohen,R.,Erez,K.,benAvraham,D.,Havlin,S.:Resilienceoftheinternettorandombreak-
downs. Phys.Rev.Lett.85,4626(2000)
10. Dorogovtsev,S.N.,Goltsev,A.V.,Mendes,J.F.F.:k-coreorganizationofcomplexnetworks.
Phys.Rev.Lett.96(4),040601(2006)
11. Fredrickson,G.H.,Andersen,H.C.:Kineticisingmodeloftheglasstransition. PhysicalRe-
viewLetters53(13),1244–1247(Sep1984),http://dx.doi.org/10.1103/PhysRevLett.53.1244
12. Go¨tze, W.: Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling The-
ory (International Series of Monographs on Physics). Oxford University Press,
USA (Feb 2009), http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-
20&path=ASIN/B007PM7EKU
13. Kitsak,M.,Gallos,L.K.,Havlin,S.,Liljeros,F.,Muchnik,L.,Stanley,H.E.,Makse,H.A.:
Identificationofinfluentialspreadersincomplexnetworks.Nat.Phys.6(11),888–893(2010)
14. Klimek, P., Thurner, S., Hanel, R.: Pruning the tree of life: k-core percolation as selection
mechanism. J.Theor.Biol.256(1),142–6(2009)
15. Schwarz,J.M.,Liu,A.J.,Chayes,L.Q.:Theonsetofjammingasthesuddenemergenceofan
infinitek-corecluster. Europhys.Lett.73,560(2006)
16. Sellitto,M.,PhysicalReviewE86,030502(R)(2012)
17. Sellitto, M., De Martino, D., Caccioli, F., Arenzon, J.J.: Dynamic facilitation picture of a
higher-orderglasssingularity. Phys.Rev.Lett.105,265704(2010)
18. Stauffer,D.,Aharony,A.:IntroductionToPercolationTheory. CRCPress,2edn.(Jul1994),
http://www.worldcat.org/isbn/0748402535