PHYSICAL REVIEW E 87 052130 (2013)

Phase transitions in supercritical explosive percolation

Wei Chen123 Jan Nagler45dagger Xueqi Cheng2 Xiaolong Jin2 Huawei Shen2 Zhiming Zheng6 and Raissa M DrsquoSouza3Dagger1School of Mathematical Sciences Peking University Beijing China

2Institute of Computing Technology Chinese Academy of Sciences Beijing China3University of California Davis California 95616 USA

4Max Planck Institute for Dynamics and Self-Organization (MPI DS) Gottingen Germany5Institute for Nonlinear Dynamics Faculty of Physics University of Gottingen Gottingen Germany

6Key Laboratory of Mathematics Informatics and Behavioral Semantics Ministry of Education Beijing University of Aeronautics andAstronautics 100191 Beijing China

(Received 9 January 2013 published 24 May 2013)

Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice orrandom network In the Bohman-Frieze-Wormald model (BFW) of percolation edges sampled from a randomgraph are considered individually and either added to the graph or rejected provided that the fraction of acceptededges is never smaller than a decreasing function with asymptotic value of α a constant The BFW processhas been studied as a model system for investigating the underlying mechanisms leading to discontinuous phasetransitions in percolation Here we focus on the regime α isin [06095] where it is known that only one giantcomponent denoted C1 initially appears at the discontinuous phase transition We show that at some pointin the supercritical regime C1 stops growing and eventually a second giant component denoted C2 emergesin a continuous percolation transition The delay between the emergence of C1 and C2 and their asymptoticsizes both depend on the value of α and we establish by several techniques that there exists a bifurcationpoint αc = 0763 plusmn 0002 For α isin [06αc) C1 stops growing the instant it emerges and the delay between theemergence of C1 and C2 decreases with increasing α For α isin (αc095] in contrast C1 continues growing intothe supercritical regime and the delay between the emergence of C1 and C2 increases with increasing α As weshow αc marks the minimal delay possible between the emergence of C1 and C2 (ie the smallest edge densityfor which C2 can exist) We also establish many features of the continuous percolation of C2 including scalingexponents and relations

DOI 101103PhysRevE87052130 PACS number(s) 6460ah 6460aq 8975Hc 0250Ey

I INTRODUCTION

Percolation in networks a phase transition from smallscattered components to large-scale connectivity is heavilystudied and widely applied in technological and social systems[1ndash4] biological networks [56] epidemiology [7ndash10] andeven dynamical models of economic systems [1112] Oneof the most classic and widely studied models is the Erdos-Renyi random graph (ER) started from a collection of n

initially isolated nodes where edges sampled uniformly fromthe complete graph are sequentially added between nodesPercolation under ER-like processes is considered a robustcontinuous phase transition with a unique giant componentemerging at the percolation threshold [13] Thus altering thelocation and nature of the percolation phase transition has beena long-standing challenge Three years ago Achlioptas et alproposed a modified ER model in which two random edgesare sampled simultaneously but only the edge connectingtwo components with smaller product of their sizes is addedwhile the other edge is discarded The authors showed thatthe percolation transition can thus be delayed considerablyand when the transition eventually happens it is extremelyabrupt calling the phenomena ldquoexplosive percolationrdquo [14]They presented strong numerical evidence suggesting that the

chenwei2012ictaccndaggerjannlddsmpgdeDaggerraissacseucdavisedu

resulting percolation transition was in fact discontinuousMany efforts were made to study similar processes on differenttopologies showing that explosive percolation is observedin scale-free networks and lattices [15ndash19] Although suchexplosive percolation phenomena resulting from the choicebetween a fixed number of random edges as studied in [14]was later demonstrated to be actually continuous [20ndash24]several alternative models are now known to exhibit trulydiscontinuous percolation transitions [25ndash37]

Chen and DrsquoSouza recently established [38] that thepercolation transition is discontinuous for a stochastic graphevolution model introduced by Bohman Frieze and Wormald[39] (the BFW model) which examines a single edge at a time(rather than a model like that in [14] which examines multipleedges at each time step) Here each edge is independentlyeither added to the graph or rejected provided that thefraction of accepted edges is never smaller than a decreasingfunction with asymptotic value of α a constant In [38]they showed that the BFW model exhibits a discontinuouspercolation transition leading to the simultaneous emergenceof multiple giant components and that the number of stablegiant components can be tuned via the parameter α In[40] the underlying mechanism leading to the discontinuousphase transition was derived namely that growth of thelargest component is dominated by overtaking when twosmaller components merge together to become the new largestcomponent Such growth by overtaking is a natural growthmechanism observed in many complex systems from ecologiesto the business world [41ndash43] Additionally the discovery of

052130-11539-3755201387(5)052130(9) copy2013 American Physical Society

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

multiple giant component is unanticipated [44] and may haveapplications in polymerization [45] network discovery [46]and epidemiology [47]

The nature of explosive percolation phase transitions hasbeen the topic of intense debate for the past few years[20ndash2448] but more recently we are learning that theevolution of such processes in the supercritical regime can alsobe unique and of great interest For instance in the supercriticalregime the size of the largest component can be a non-self-averaging quantity [49] furthermore there can be multiplediscontinuous jumps in the size of the largest component [50]Here we investigate both numerically and analytically ageneralized BFW model in the supercritical regime

Specifically we study the initial percolation transition andsupercritical evolution of the BFW model for the parameterα isin [06095] (This regime is simplest as only one giantcomponent initially emerges in a discontinuous phase tran-sition) We show that the largest component eventually stopsgrowing leading to the emergence of a second giant componentat some delayed time and that the second component isstable persisting throughout the subsequent evolution Viaan extensive finite size scaling analysis we establish that thesecond transition is continuous The main result we derive isthat there exists a critical value αc such that for α isin [06αc)

C1 stops growing at the discontinuous percolation transi-tion yet for α isin (αc095] C1 continues growing for sometime into the supercritical regime see Fig 1(a) Furthermorefor α lt αc the value of C1 at percolation threshold becomessmaller with increasing α As fewer edges are necessary tocreate a smaller C1 this leads to enhanced growth of C2

in the supercritical regime Thus the gap between the twopercolation transitions is smaller with increasing α for α lt αcFor α gt αc in contrast the value of C1 at the time when itstops growing in the supercritical regime becomes larger withincreasing α As more edges are required to create a largerC1 this leads to delayed growth of C2 Thus the gap betweenthe two percolation transitions is larger with increasing α forα gt αc As we show αc marks the minimal delay possiblebetween the emergence of C1 and C2 (ie the smallest edgedensity for which C2 can exist)

The rest of the paper is organized as follows In Sec IIwe discuss the BFW model and show further evidence for thediscontinuous transition of the largest component In Sec IIIwe analyze the growth cessation of the largest component inthe supercritical regime Section IV contains an analysis of thecritical behavior of the second phase transition through finitesize scaling We conclude with some discussion in Sec V

II THE BFW MODEL AND THE DISCONTINUOUSTRANSITION IN C1

In this section we illustrate the BFW model The system isinitialized with N isolated nodes and a cap k on the maximumallowed component size set to k = 2 and increased in stages asfollows Edges are sampled one at a time uniformly at randomfrom the complete graph If an edge would lead to formationof a component of size less than or equal to k it is acceptedOtherwise the edge is rejected provided that the fraction ofaccepted edges remains greater than or equal to a functiong(k) = α + (2k)minus12 where α is a tunable parameter (In the

06 07 08 09 1 11 120

02

04

06

08

1

p

C1

(a)

α=06

α=07

α=075

α=08

α=09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

005

01

015

02

p

C2

(b)

α=06

α=07

α=075

α=08

α=09

FIG 1 (Color online) Typical evolution of the BFW model(a) Fraction of nodes in the largest component C1 = |C1|N asa function of edge density p equiv tN for different values of αshowing a discontinuous phase transition at a location dependenton the value of α As derived herein for α gt αc the size of C1

increases in the supercritical regime (b) Fraction of nodes in thesecond largest component C2 = |C2|N as a function of p for α =07508070609 from left to right showing an instantaneous peakwhen C1 emerges followed later by a continuous phase transition Theextent of delay between the discontinuous emergence of C1 and thecontinuous emergence of C2 depends in the value of α The systemsize is N = 106

original BFW model Bohman Frieze and Wormald onlystudied the case of α = 12 and showed that the percolationthreshold of this model is larger than 0966 89) Finally ifthe fraction of accepted nodes would drop below g(k) thecap is augmented to k + 1 and the impact of adding theedge re-evaluated against the new values k + 1 and g(k + 1)This final step of augmenting the cap and re-evaluating theimpact is iterated until either k increases sufficiently to acceptthe edge or g(k) decreases sufficiently that the edge can berejected Asymptotically the fraction of accepted edges islimkrarrinfin g(k) = α

Stating the BFW algorithm in detail requires some notationLet k denote the cap size N denote the number of nodes u

denote the total number of edges sampled A denote the setof accepted edges and t = |A| denote the number of acceptededges At each step u an edge eu is sampled uniformly atrandom from the complete graph generated by the N nodes

052130-2

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

and evaluated by the following algorithm stopping once aspecified number of edges ωN have been accepted

Repeat until u = ωN

Set l = size of largest component in A cup euif (l k) A larr A cup euu larr u + 1 else if (tu lt g(k))k larr k + 1 else u larr u + 1 It has been shown in Ref [38] that the percolation transition

of the order parameter |C1|N = C1 defined as the fractionof nodes in the largest component is discontinuous in thethermodynamic limit for α isin (0097] [see eg Fig 1(a)]This holds regardless of whether we sample uniformly atrandom all edges from the complete graph or sample onlyedges not yet existing in the graph [38] (The former allows

105

106

107

02

025

03

035

04

045

N

Δ C

max

(a)

105

106

107

06

07

08

09

1

N

Pinfin

(b)

105

106

107

004

006

01

02

N

Mrsquo 2

(c)

FIG 2 (Color online) (a) The largest jump in the size of C1

resulting from one edge as a function of system sizes for α =06070809095 ordered from top to bottom (b) The orderparameter Pinfin which is C1 immediately after the largest jumpof C1 as a function of system sizes for α = 06070809095from top to bottom (c) The maximum of the second moment ofthe cluster-size distribution M prime

2 as a function of system size forα = 06070809095 from top to bottom All the quantities in(a) (b) and (c) are asymptotically independent of system sizes andthe solid lines are the mean values These results indicate that thepercolation transitions are discontinuous Each datum is averagedover 500 realizations

TABLE I Summary of numerical results showing discontinuousphase transitions in the order parameter C1

α Cmax Pinfin M prime2

06 0391 plusmn 0005 0947 plusmn 0004 0154 plusmn 000507 0374 plusmn 0006 0901 plusmn 0005 0139 plusmn 000608 0344 plusmn 0005 0829 plusmn 0004 0118 plusmn 000609 0291 plusmn 0005 0702 plusmn 0005 0084 plusmn 0006095 0240 plusmn 0006 0582 plusmn 0005 0057 plusmn 0008

intracomponent edges including multiple edges and self-edges) Here we provide additional numerical evidence forthe discontinuous transition by examining three quantities thelargest jump in C1 resulting from adding one edge denotedas Cmax order parameter C1 immediately after the largestjump denoted as Pinfin and the maximum in the second momentof the relative-size distribution of components excluding thecontribution of the largest component

M prime2 = M2 minus C2

1 (1)

where M2 = sumi C

2i and Ci is the relative size of component

i If CmaxPinfinM prime2 converge to some positive constant in

the thermodynamic limit the transition is discontinuous whilethey vanish for continuous phase transitions [21262837] Weobserve CmaxPinfin and M prime

2 are all asymptotically indepen-dent of system sizes within error bars which are shown inFigs 2(a)ndash2(c) respectively For large size systems and differ-ing values of α we obtain the asymptotic values of Cmax Pinfinand M prime

2 for α = 06070809095 respectively (Table I)These numerical results indicate that the percolation transitionis discontinuous for the range studied here α isin [06095]

To estimate the percolation threshold we implementthe numerical methods proposed in Refs [2837] Wemeasure the edge density pc1(N ) at which the largestjump in the size of C1 occurs and the edge den-sity pc2(N ) at which M prime

2 attains maximum for differentsystem sizes N Extrapolating these estimators in ther-modynamic limit we obtain pc = 0948 plusmn 00070915 plusmn00080862 plusmn 00070780 plusmn 00070711 plusmn 0006 for α =06070809095 respectively These results indicate thepercolation threshold for the emergence of a giant compo-nent is larger for smaller α We further observe asymp-totic power law relation between |pci minus pc|(i = 12) andsystem size N In particular |pci minus pc| sim Nminusδ with δ =03 plusmn 0104 plusmn 0104 plusmn 0105 plusmn 0105 plusmn 01 for α =06070809095 respectively The exponent δ = 05 plusmn01 for α = 09095 is the same as what is observed forBFW model (α = 05) on a square lattice within errorbars [37]

III GROWTH CESSATION OF THE LARGESTCOMPONENT

The discontinuous transition of the order parameter C1

has been substantiated [38] and the underlying mechanismaccounting for the discontinuous transition has been identified[40] However the behavior of C1 after the largest jump (ie insupercritical regime) has not been studied previously Here we

052130-3

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

study C1 for values of the edge density p isin [010] for differentsystem sizes Interestingly we observe that C1 stops increasingeither immediately after the largest jump of C1 or at somedelayed point dependent on the specific α isin [06095] Thisgrowth cessation of the largest component in the supercriticalregime has however not been reported in other models inwhich any edge from the complete graph is allowed to besampled

Using the notation in Sec II t denotes the number ofaccepted edges and thus tN is the edge density (also denotedthroughout by the shorthand p equiv tN) Let us denote tc asthe point where the largest jump of C1 occurs and kc as thevalue of stage k at tc Also denote kfinal as the final valuewhen k stops increasing and t(kfinal) as the point where k stopsincreasing We measure k(tc)kfinaltcNt(kfinal)N for differ-ent α in a system of size N = 106 Figure 3(a) shows whenα lt αc asymp 0763 k(tc) and kfinal overlap and both decreasewith α When α gt αc however k(tc) keeps on decreasingwhile kfinal increases with α Therefore kfinalN reachesa minimum at α = αc Let kN = (kfinal minus k(tc))N andFig 3(b) shows kN sim Nminus107Nminus093 for α = 076075respectively indicating kn rarr 0 as n rarr infin For α = 077

however kN converges to some positive constant Moreprecise investigation detailed below shows the transition pointαc = 0763 plusmn 0002 Figure 3(c) shows tcN and t(kfinal)Noverlap when α lt αc asymp 0763 and there is a finite gap betweenthem once α gt αc We observe that t(kfinal) reaches a minimumat αc Let tN = (t(kfinal) minus tc)N and in Fig 3(d) tN

is plotted as a function of N for α = 075076077 wheretN sim Nminus103Nminus082 for α = 076075 respectively butasymptotically converges to some positive constant for α =077 This numerical result shows the same transition point atαc asymp 0763

To understand the underlying mechanism accounting for thefact that the stage k and the size of the largest component stopsincreasing in the supercritical regime it is useful to measureP (ktN ) defined as the probability of sampling a randomedge that keeps C1N k if it is added at step t in a systemof size N If limNrarrinfin P (ktN ) limkrarrinfin g(k) = α alwaysholds for t gt tc where k sim O(N ) the fraction of acceptededges over total sampled edges tu quickly converges to somevalue larger than α so we can always discard an edge thatwould increase C1 thus stage stops increasing However iflimNrarrinfin P (ktN ) lt α the stage k keeps on increasing until

06 07 08 0905

06

07

08

09

1

α

αcasymp0763

(a)

k(tc)N

kfinal

N

y=(1+(2αminus1)12)2

06 07 08 090

1

2

3

4

5

6

7

α

αcasymp0763

(c)tcN

t(kfinal

)N

105

106

10minus6

10minus4

10minus2

100

102

N

Δ tN

minus082

minus103

(d)

α=077α=076α=075

105

106

10minus6

10minus4

10minus2

100

N

Δ k

N

minus093

minus107

(b)

α=077α=076α=075

FIG 3 (Color online) System size is N = 106 for (a)(c) (a) Behavior of the cap k versus α specifically measuring k(tc)N (blue circles)the value at the discontinuous emergence of C1 and kfinalN (red triangles) the value when k stops increasing Both lines intersect the functionderived in Eq (4) y = (1 + (2α minus 1)12)2 at the same point αc asymp 0763 (b) kN versus system size for α = 075076077 (orderedbottom to top) showing kN converges to 0 for α = 075076 while it converges to a positive constant for α = 077 Further analysis showsα = 0763 plusmn 0002 is the transition point between the behaviors (c) tcN and t(kfinal)N as a function of α (d) tN versus system size forα = 075076077 (ordered bottom to top) showing tN converges to 0 asymptotically for α = 075076 while it asymptotically convergesto some positive constant for α = 077 More detailed analysis shows α = 0763 plusmn 0002 is the transition point between the two behaviors

052130-4

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

the following condition holds at t = t(kfinal)

limNrarrinfin

P (ktN ) = α (2)

We measure k(tc)N and kfinalN with system sizes N = 106

for over 100 realizations We find that at the point where thelargest jump of C1 occurs namely tcN for all α isin [06095]k(tc)N = C1 gt 05 averaged over 1000 realizations Dueto this fact once t gt tc the growth of C1 can only occurdue to the mechanism of direct growth [21] in particularthe edge connecting the largest component and anothersmaller component results in the growth of C1 Thereforethe probability of sampling a random edge which leads to thegrowth of C1 is 2(1 minus C1)C1 and thus P (ktN ) satisfies

limNrarrinfin

P (ktN ) = 1 minus 2(1 minus C1)C1 (3)

Combining Eqs (2) and (3) we obtain C1 at t(kfinal) for infinitesystem

C1 = 1 + radic2α minus 1

2 (4)

Figure 3(a) shows the dashed line of Eq (4) intersects k(tc)Nand kfinalN at the same point αc = 0763 within error bars

for system size N = 106 For α lt αc k(tc)N gt 1+radic2αminus12

(blue circles lie above the dashed line) and P (ktcN ) gt αTherefore the stage k stops increasing immediately after thelargest jump of C1 and kfinalN is almost the same as k(tc)N(blue circles and red triangles overlap) However for α gt αck(tc)N lt 1+radic

2αminus12 (blue circles lie below the dashed line)

Therefore both the stage k and C1 increase until Eq (4) holds(red triangles and the dashed line overlap)

IV CONTINUOUS TRANSITION OF THE SECONDLARGEST COMPONENT

The size of the largest component stops increasing at thepoint t(kfinal)N (which is either at the discontinuous phasetransition point or in the supercritical regime dependent onthe value of α) Yet the second largest component continuesgrowing at this point Two natural and interesting questionsfollow Is there a second giant component emerging at somestep t gt t(kfinal) If such percolation transition occurs is ita discontinuous phase transition as we find for the orderparameter C1 To answer these questions we first numericallymeasure C2 as a function of the edge density p equiv tN for

35 37 39 41 43 4510

minus4

10minus3

10minus2

10minus1

p

C2

α=07(a)

32 34 36 38 4 4210

minus4

10minus3

10minus2

10minus1

p

C2

α=08(g)

3 32 34 36 38 410

minus2

10minus1

100

p

C2N

βν

α=08(h)

22 24 26 28 3 3210

minus4

10minus3

10minus2

10minus1

p

C2

α=0763(d)

minus10 minus5 0 5 10006

01

02

03

04

(pminuspcrsquo)N1ν

C2N

βν

α=0763(f)

minus10 minus5 0 5 10002

004

006

01

02

(pminuspcrsquo)N1ν

C2N

βν

α=08(i)

35 37 39 41 43 4510

minus2

10minus1

100

p

C2N

βν

α=07

(b)

minus10 minus5 0 5 10006

008

01

02

03

(pminuspcrsquo)N1ν

C2N

βν

α=07(c)

23 25 27 29 31 3310

minus2

10minus1

100

p

C2N

βν

α=0763

(e)

FIG 4 (Color online) In (a) (d) (g) fraction of nodes in the second largest component C2 is plotted as a function of edge density p forα = 07076308 respectively (b) (e) (h) show their rescaling C2N

βν respectively (c) (f) (i) show C2Nβν versus (p minus pprime

c)N1ν indicating

the validity of Eq (7) The system size goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-5

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

α = 06065070750808509 As shown in Fig 1(b)we see C2 (the fractional size of component C2) spikeinstantaneously at the first discontinuous transition then itdisappears and we observe a seemingly continuous percolationtransition when C2 again emerges at some point in thesupercritical regime

To investigate the continuous nature of percolation transi-tion of C2 we make use of finite size scaling [51] If a phasetransition is continuous every variable X is believed to bescale independent and obeys the following finite size scalingform near the percolation threshold pc

X = NminusθνF [(p minus pc)N1ν]

where θ and ν are critical exponents N is the system sizeand F is a universal function If the dimension of the system d

exceeds the upper critical dimension dc the finite-size scalingform for a variable X is assumed to satisfy

X = NminusθνF [(p minus pc)N1ν] (5)

where ν = dcνm and νm is the mean-field value of ν As aresult X follows a power law at p = pc X sim Nminusθν Random

networks display a mean-filed type critical behavior [52]and lack any space dimensionality essentially d = infin Herewe consider two variables in the percolation process thefraction of nodes in the second largest component C2 and thesusceptibility χ The susceptibility χ is defined as the standarddeviation of the size of the second largest component

χ = N

radiclangC2

2rang minus 〈C2〉2 (6)

We assume that C2 and χ obey the following scaling relations

C2 = NminusβνF (1)[(p minus pprimec)N1ν] (7)

χ = NγνF (2)[(p minus pprimec)N1ν] (8)

where F (1) and F (2) are different universal functions (howeverthey are related to each other) pprime

c denotes the percolationthreshold of the second phase transition and βγν are criticalexponents characterizing the transition We can deduce fromthe definition of χ [Eq (6)] and the scaling relations inEqs (7) and (8) at p = pprime

c

βν = 1 minus γ ν (9)

35 4 45 5 550

0002

0004

0006

0008

001

p

χN

α=07(a)

3 35 4 45 50

0002

0004

0006

0008

001

p

χN

α=08(c)

25 3 35 40

0002

0004

0006

0008

001

p

χN

α=0763(b)

104

105

106

107

10minus3

10minus2

N

y=minus0307xminus2277

y=minus0323xminus2602

α=07(d)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0254xminus2606

α=0763

y=minus0251xminus3197

(e)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0237xminus2821

α=08

y=minus0231xminus3416

(f) C2(p

crsquo)

χ(pcrsquo)N

FIG 5 (Color online) In (a) (b) (c) χN is plotted as a function of p for α = 07076308 respectively showing the peak of χN

moves leftward as system size increases The dashed lines are the percolation threshold obtained from numerical simulation through Eq (8)The peaks of χN is moving leftward to the percolation threshold pprime

c which indicates the continuous nature of the second phase transition In(d) (e) (f) C2 and χN at the percolation threshold pprime

c is plotted as a function of system sizes for α = 07076308 respectively The systemsize goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-6

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

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CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

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PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

and evaluated by the following algorithm stopping once aspecified number of edges ωN have been accepted

Repeat until u = ωN

Set l = size of largest component in A cup euif (l k) A larr A cup euu larr u + 1 else if (tu lt g(k))k larr k + 1 else u larr u + 1 It has been shown in Ref [38] that the percolation transition

of the order parameter |C1|N = C1 defined as the fractionof nodes in the largest component is discontinuous in thethermodynamic limit for α isin (0097] [see eg Fig 1(a)]This holds regardless of whether we sample uniformly atrandom all edges from the complete graph or sample onlyedges not yet existing in the graph [38] (The former allows

105

106

107

02

025

03

035

04

045

N

Δ C

max

(a)

105

106

107

06

07

08

09

1

N

Pinfin

(b)

105

106

107

004

006

01

02

N

Mrsquo 2

(c)

FIG 2 (Color online) (a) The largest jump in the size of C1

resulting from one edge as a function of system sizes for α =06070809095 ordered from top to bottom (b) The orderparameter Pinfin which is C1 immediately after the largest jumpof C1 as a function of system sizes for α = 06070809095from top to bottom (c) The maximum of the second moment ofthe cluster-size distribution M prime

2 as a function of system size forα = 06070809095 from top to bottom All the quantities in(a) (b) and (c) are asymptotically independent of system sizes andthe solid lines are the mean values These results indicate that thepercolation transitions are discontinuous Each datum is averagedover 500 realizations

TABLE I Summary of numerical results showing discontinuousphase transitions in the order parameter C1

α Cmax Pinfin M prime2

06 0391 plusmn 0005 0947 plusmn 0004 0154 plusmn 000507 0374 plusmn 0006 0901 plusmn 0005 0139 plusmn 000608 0344 plusmn 0005 0829 plusmn 0004 0118 plusmn 000609 0291 plusmn 0005 0702 plusmn 0005 0084 plusmn 0006095 0240 plusmn 0006 0582 plusmn 0005 0057 plusmn 0008

intracomponent edges including multiple edges and self-edges) Here we provide additional numerical evidence forthe discontinuous transition by examining three quantities thelargest jump in C1 resulting from adding one edge denotedas Cmax order parameter C1 immediately after the largestjump denoted as Pinfin and the maximum in the second momentof the relative-size distribution of components excluding thecontribution of the largest component

M prime2 = M2 minus C2

1 (1)

where M2 = sumi C

2i and Ci is the relative size of component

i If CmaxPinfinM prime2 converge to some positive constant in

the thermodynamic limit the transition is discontinuous whilethey vanish for continuous phase transitions [21262837] Weobserve CmaxPinfin and M prime

2 are all asymptotically indepen-dent of system sizes within error bars which are shown inFigs 2(a)ndash2(c) respectively For large size systems and differ-ing values of α we obtain the asymptotic values of Cmax Pinfinand M prime

2 for α = 06070809095 respectively (Table I)These numerical results indicate that the percolation transitionis discontinuous for the range studied here α isin [06095]

To estimate the percolation threshold we implementthe numerical methods proposed in Refs [2837] Wemeasure the edge density pc1(N ) at which the largestjump in the size of C1 occurs and the edge den-sity pc2(N ) at which M prime

2 attains maximum for differentsystem sizes N Extrapolating these estimators in ther-modynamic limit we obtain pc = 0948 plusmn 00070915 plusmn00080862 plusmn 00070780 plusmn 00070711 plusmn 0006 for α =06070809095 respectively These results indicate thepercolation threshold for the emergence of a giant compo-nent is larger for smaller α We further observe asymp-totic power law relation between |pci minus pc|(i = 12) andsystem size N In particular |pci minus pc| sim Nminusδ with δ =03 plusmn 0104 plusmn 0104 plusmn 0105 plusmn 0105 plusmn 01 for α =06070809095 respectively The exponent δ = 05 plusmn01 for α = 09095 is the same as what is observed forBFW model (α = 05) on a square lattice within errorbars [37]

III GROWTH CESSATION OF THE LARGESTCOMPONENT

The discontinuous transition of the order parameter C1

has been substantiated [38] and the underlying mechanismaccounting for the discontinuous transition has been identified[40] However the behavior of C1 after the largest jump (ie insupercritical regime) has not been studied previously Here we

052130-3

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

study C1 for values of the edge density p isin [010] for differentsystem sizes Interestingly we observe that C1 stops increasingeither immediately after the largest jump of C1 or at somedelayed point dependent on the specific α isin [06095] Thisgrowth cessation of the largest component in the supercriticalregime has however not been reported in other models inwhich any edge from the complete graph is allowed to besampled

Using the notation in Sec II t denotes the number ofaccepted edges and thus tN is the edge density (also denotedthroughout by the shorthand p equiv tN) Let us denote tc asthe point where the largest jump of C1 occurs and kc as thevalue of stage k at tc Also denote kfinal as the final valuewhen k stops increasing and t(kfinal) as the point where k stopsincreasing We measure k(tc)kfinaltcNt(kfinal)N for differ-ent α in a system of size N = 106 Figure 3(a) shows whenα lt αc asymp 0763 k(tc) and kfinal overlap and both decreasewith α When α gt αc however k(tc) keeps on decreasingwhile kfinal increases with α Therefore kfinalN reachesa minimum at α = αc Let kN = (kfinal minus k(tc))N andFig 3(b) shows kN sim Nminus107Nminus093 for α = 076075respectively indicating kn rarr 0 as n rarr infin For α = 077

however kN converges to some positive constant Moreprecise investigation detailed below shows the transition pointαc = 0763 plusmn 0002 Figure 3(c) shows tcN and t(kfinal)Noverlap when α lt αc asymp 0763 and there is a finite gap betweenthem once α gt αc We observe that t(kfinal) reaches a minimumat αc Let tN = (t(kfinal) minus tc)N and in Fig 3(d) tN

is plotted as a function of N for α = 075076077 wheretN sim Nminus103Nminus082 for α = 076075 respectively butasymptotically converges to some positive constant for α =077 This numerical result shows the same transition point atαc asymp 0763

To understand the underlying mechanism accounting for thefact that the stage k and the size of the largest component stopsincreasing in the supercritical regime it is useful to measureP (ktN ) defined as the probability of sampling a randomedge that keeps C1N k if it is added at step t in a systemof size N If limNrarrinfin P (ktN ) limkrarrinfin g(k) = α alwaysholds for t gt tc where k sim O(N ) the fraction of acceptededges over total sampled edges tu quickly converges to somevalue larger than α so we can always discard an edge thatwould increase C1 thus stage stops increasing However iflimNrarrinfin P (ktN ) lt α the stage k keeps on increasing until

06 07 08 0905

06

07

08

09

1

α

αcasymp0763

(a)

k(tc)N

kfinal

N

y=(1+(2αminus1)12)2

06 07 08 090

1

2

3

4

5

6

7

α

αcasymp0763

(c)tcN

t(kfinal

)N

105

106

10minus6

10minus4

10minus2

100

102

N

Δ tN

minus082

minus103

(d)

α=077α=076α=075

105

106

10minus6

10minus4

10minus2

100

N

Δ k

N

minus093

minus107

(b)

α=077α=076α=075

FIG 3 (Color online) System size is N = 106 for (a)(c) (a) Behavior of the cap k versus α specifically measuring k(tc)N (blue circles)the value at the discontinuous emergence of C1 and kfinalN (red triangles) the value when k stops increasing Both lines intersect the functionderived in Eq (4) y = (1 + (2α minus 1)12)2 at the same point αc asymp 0763 (b) kN versus system size for α = 075076077 (orderedbottom to top) showing kN converges to 0 for α = 075076 while it converges to a positive constant for α = 077 Further analysis showsα = 0763 plusmn 0002 is the transition point between the behaviors (c) tcN and t(kfinal)N as a function of α (d) tN versus system size forα = 075076077 (ordered bottom to top) showing tN converges to 0 asymptotically for α = 075076 while it asymptotically convergesto some positive constant for α = 077 More detailed analysis shows α = 0763 plusmn 0002 is the transition point between the two behaviors

052130-4

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

the following condition holds at t = t(kfinal)

limNrarrinfin

P (ktN ) = α (2)

We measure k(tc)N and kfinalN with system sizes N = 106

for over 100 realizations We find that at the point where thelargest jump of C1 occurs namely tcN for all α isin [06095]k(tc)N = C1 gt 05 averaged over 1000 realizations Dueto this fact once t gt tc the growth of C1 can only occurdue to the mechanism of direct growth [21] in particularthe edge connecting the largest component and anothersmaller component results in the growth of C1 Thereforethe probability of sampling a random edge which leads to thegrowth of C1 is 2(1 minus C1)C1 and thus P (ktN ) satisfies

limNrarrinfin

P (ktN ) = 1 minus 2(1 minus C1)C1 (3)

Combining Eqs (2) and (3) we obtain C1 at t(kfinal) for infinitesystem

C1 = 1 + radic2α minus 1

2 (4)

Figure 3(a) shows the dashed line of Eq (4) intersects k(tc)Nand kfinalN at the same point αc = 0763 within error bars

for system size N = 106 For α lt αc k(tc)N gt 1+radic2αminus12

(blue circles lie above the dashed line) and P (ktcN ) gt αTherefore the stage k stops increasing immediately after thelargest jump of C1 and kfinalN is almost the same as k(tc)N(blue circles and red triangles overlap) However for α gt αck(tc)N lt 1+radic

2αminus12 (blue circles lie below the dashed line)

Therefore both the stage k and C1 increase until Eq (4) holds(red triangles and the dashed line overlap)

IV CONTINUOUS TRANSITION OF THE SECONDLARGEST COMPONENT

The size of the largest component stops increasing at thepoint t(kfinal)N (which is either at the discontinuous phasetransition point or in the supercritical regime dependent onthe value of α) Yet the second largest component continuesgrowing at this point Two natural and interesting questionsfollow Is there a second giant component emerging at somestep t gt t(kfinal) If such percolation transition occurs is ita discontinuous phase transition as we find for the orderparameter C1 To answer these questions we first numericallymeasure C2 as a function of the edge density p equiv tN for

35 37 39 41 43 4510

minus4

10minus3

10minus2

10minus1

p

C2

α=07(a)

32 34 36 38 4 4210

minus4

10minus3

10minus2

10minus1

p

C2

α=08(g)

3 32 34 36 38 410

minus2

10minus1

100

p

C2N

βν

α=08(h)

22 24 26 28 3 3210

minus4

10minus3

10minus2

10minus1

p

C2

α=0763(d)

minus10 minus5 0 5 10006

01

02

03

04

(pminuspcrsquo)N1ν

C2N

βν

α=0763(f)

minus10 minus5 0 5 10002

004

006

01

02

(pminuspcrsquo)N1ν

C2N

βν

α=08(i)

35 37 39 41 43 4510

minus2

10minus1

100

p

C2N

βν

α=07

(b)

minus10 minus5 0 5 10006

008

01

02

03

(pminuspcrsquo)N1ν

C2N

βν

α=07(c)

23 25 27 29 31 3310

minus2

10minus1

100

p

C2N

βν

α=0763

(e)

FIG 4 (Color online) In (a) (d) (g) fraction of nodes in the second largest component C2 is plotted as a function of edge density p forα = 07076308 respectively (b) (e) (h) show their rescaling C2N

βν respectively (c) (f) (i) show C2Nβν versus (p minus pprime

c)N1ν indicating

the validity of Eq (7) The system size goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-5

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

α = 06065070750808509 As shown in Fig 1(b)we see C2 (the fractional size of component C2) spikeinstantaneously at the first discontinuous transition then itdisappears and we observe a seemingly continuous percolationtransition when C2 again emerges at some point in thesupercritical regime

To investigate the continuous nature of percolation transi-tion of C2 we make use of finite size scaling [51] If a phasetransition is continuous every variable X is believed to bescale independent and obeys the following finite size scalingform near the percolation threshold pc

X = NminusθνF [(p minus pc)N1ν]

where θ and ν are critical exponents N is the system sizeand F is a universal function If the dimension of the system d

exceeds the upper critical dimension dc the finite-size scalingform for a variable X is assumed to satisfy

X = NminusθνF [(p minus pc)N1ν] (5)

where ν = dcνm and νm is the mean-field value of ν As aresult X follows a power law at p = pc X sim Nminusθν Random

networks display a mean-filed type critical behavior [52]and lack any space dimensionality essentially d = infin Herewe consider two variables in the percolation process thefraction of nodes in the second largest component C2 and thesusceptibility χ The susceptibility χ is defined as the standarddeviation of the size of the second largest component

χ = N

radiclangC2

2rang minus 〈C2〉2 (6)

We assume that C2 and χ obey the following scaling relations

C2 = NminusβνF (1)[(p minus pprimec)N1ν] (7)

χ = NγνF (2)[(p minus pprimec)N1ν] (8)

where F (1) and F (2) are different universal functions (howeverthey are related to each other) pprime

c denotes the percolationthreshold of the second phase transition and βγν are criticalexponents characterizing the transition We can deduce fromthe definition of χ [Eq (6)] and the scaling relations inEqs (7) and (8) at p = pprime

c

βν = 1 minus γ ν (9)

35 4 45 5 550

0002

0004

0006

0008

001

p

χN

α=07(a)

3 35 4 45 50

0002

0004

0006

0008

001

p

χN

α=08(c)

25 3 35 40

0002

0004

0006

0008

001

p

χN

α=0763(b)

104

105

106

107

10minus3

10minus2

N

y=minus0307xminus2277

y=minus0323xminus2602

α=07(d)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0254xminus2606

α=0763

y=minus0251xminus3197

(e)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0237xminus2821

α=08

y=minus0231xminus3416

(f) C2(p

crsquo)

χ(pcrsquo)N

FIG 5 (Color online) In (a) (b) (c) χN is plotted as a function of p for α = 07076308 respectively showing the peak of χN

moves leftward as system size increases The dashed lines are the percolation threshold obtained from numerical simulation through Eq (8)The peaks of χN is moving leftward to the percolation threshold pprime

c which indicates the continuous nature of the second phase transition In(d) (e) (f) C2 and χN at the percolation threshold pprime

c is plotted as a function of system sizes for α = 07076308 respectively The systemsize goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-6

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

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CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

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PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

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CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

study C1 for values of the edge density p isin [010] for differentsystem sizes Interestingly we observe that C1 stops increasingeither immediately after the largest jump of C1 or at somedelayed point dependent on the specific α isin [06095] Thisgrowth cessation of the largest component in the supercriticalregime has however not been reported in other models inwhich any edge from the complete graph is allowed to besampled

Using the notation in Sec II t denotes the number ofaccepted edges and thus tN is the edge density (also denotedthroughout by the shorthand p equiv tN) Let us denote tc asthe point where the largest jump of C1 occurs and kc as thevalue of stage k at tc Also denote kfinal as the final valuewhen k stops increasing and t(kfinal) as the point where k stopsincreasing We measure k(tc)kfinaltcNt(kfinal)N for differ-ent α in a system of size N = 106 Figure 3(a) shows whenα lt αc asymp 0763 k(tc) and kfinal overlap and both decreasewith α When α gt αc however k(tc) keeps on decreasingwhile kfinal increases with α Therefore kfinalN reachesa minimum at α = αc Let kN = (kfinal minus k(tc))N andFig 3(b) shows kN sim Nminus107Nminus093 for α = 076075respectively indicating kn rarr 0 as n rarr infin For α = 077

however kN converges to some positive constant Moreprecise investigation detailed below shows the transition pointαc = 0763 plusmn 0002 Figure 3(c) shows tcN and t(kfinal)Noverlap when α lt αc asymp 0763 and there is a finite gap betweenthem once α gt αc We observe that t(kfinal) reaches a minimumat αc Let tN = (t(kfinal) minus tc)N and in Fig 3(d) tN

is plotted as a function of N for α = 075076077 wheretN sim Nminus103Nminus082 for α = 076075 respectively butasymptotically converges to some positive constant for α =077 This numerical result shows the same transition point atαc asymp 0763

To understand the underlying mechanism accounting for thefact that the stage k and the size of the largest component stopsincreasing in the supercritical regime it is useful to measureP (ktN ) defined as the probability of sampling a randomedge that keeps C1N k if it is added at step t in a systemof size N If limNrarrinfin P (ktN ) limkrarrinfin g(k) = α alwaysholds for t gt tc where k sim O(N ) the fraction of acceptededges over total sampled edges tu quickly converges to somevalue larger than α so we can always discard an edge thatwould increase C1 thus stage stops increasing However iflimNrarrinfin P (ktN ) lt α the stage k keeps on increasing until

06 07 08 0905

06

07

08

09

1

α

αcasymp0763

(a)

k(tc)N

kfinal

N

y=(1+(2αminus1)12)2

06 07 08 090

1

2

3

4

5

6

7

α

αcasymp0763

(c)tcN

t(kfinal

)N

105

106

10minus6

10minus4

10minus2

100

102

N

Δ tN

minus082

minus103

(d)

α=077α=076α=075

105

106

10minus6

10minus4

10minus2

100

N

Δ k

N

minus093

minus107

(b)

α=077α=076α=075

FIG 3 (Color online) System size is N = 106 for (a)(c) (a) Behavior of the cap k versus α specifically measuring k(tc)N (blue circles)the value at the discontinuous emergence of C1 and kfinalN (red triangles) the value when k stops increasing Both lines intersect the functionderived in Eq (4) y = (1 + (2α minus 1)12)2 at the same point αc asymp 0763 (b) kN versus system size for α = 075076077 (orderedbottom to top) showing kN converges to 0 for α = 075076 while it converges to a positive constant for α = 077 Further analysis showsα = 0763 plusmn 0002 is the transition point between the behaviors (c) tcN and t(kfinal)N as a function of α (d) tN versus system size forα = 075076077 (ordered bottom to top) showing tN converges to 0 asymptotically for α = 075076 while it asymptotically convergesto some positive constant for α = 077 More detailed analysis shows α = 0763 plusmn 0002 is the transition point between the two behaviors

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PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

the following condition holds at t = t(kfinal)

limNrarrinfin

P (ktN ) = α (2)

We measure k(tc)N and kfinalN with system sizes N = 106

for over 100 realizations We find that at the point where thelargest jump of C1 occurs namely tcN for all α isin [06095]k(tc)N = C1 gt 05 averaged over 1000 realizations Dueto this fact once t gt tc the growth of C1 can only occurdue to the mechanism of direct growth [21] in particularthe edge connecting the largest component and anothersmaller component results in the growth of C1 Thereforethe probability of sampling a random edge which leads to thegrowth of C1 is 2(1 minus C1)C1 and thus P (ktN ) satisfies

limNrarrinfin

P (ktN ) = 1 minus 2(1 minus C1)C1 (3)

Combining Eqs (2) and (3) we obtain C1 at t(kfinal) for infinitesystem

C1 = 1 + radic2α minus 1

2 (4)

Figure 3(a) shows the dashed line of Eq (4) intersects k(tc)Nand kfinalN at the same point αc = 0763 within error bars

for system size N = 106 For α lt αc k(tc)N gt 1+radic2αminus12

(blue circles lie above the dashed line) and P (ktcN ) gt αTherefore the stage k stops increasing immediately after thelargest jump of C1 and kfinalN is almost the same as k(tc)N(blue circles and red triangles overlap) However for α gt αck(tc)N lt 1+radic

2αminus12 (blue circles lie below the dashed line)

Therefore both the stage k and C1 increase until Eq (4) holds(red triangles and the dashed line overlap)

IV CONTINUOUS TRANSITION OF THE SECONDLARGEST COMPONENT

The size of the largest component stops increasing at thepoint t(kfinal)N (which is either at the discontinuous phasetransition point or in the supercritical regime dependent onthe value of α) Yet the second largest component continuesgrowing at this point Two natural and interesting questionsfollow Is there a second giant component emerging at somestep t gt t(kfinal) If such percolation transition occurs is ita discontinuous phase transition as we find for the orderparameter C1 To answer these questions we first numericallymeasure C2 as a function of the edge density p equiv tN for

35 37 39 41 43 4510

minus4

10minus3

10minus2

10minus1

p

C2

α=07(a)

32 34 36 38 4 4210

minus4

10minus3

10minus2

10minus1

p

C2

α=08(g)

3 32 34 36 38 410

minus2

10minus1

100

p

C2N

βν

α=08(h)

22 24 26 28 3 3210

minus4

10minus3

10minus2

10minus1

p

C2

α=0763(d)

minus10 minus5 0 5 10006

01

02

03

04

(pminuspcrsquo)N1ν

C2N

βν

α=0763(f)

minus10 minus5 0 5 10002

004

006

01

02

(pminuspcrsquo)N1ν

C2N

βν

α=08(i)

35 37 39 41 43 4510

minus2

10minus1

100

p

C2N

βν

α=07

(b)

minus10 minus5 0 5 10006

008

01

02

03

(pminuspcrsquo)N1ν

C2N

βν

α=07(c)

23 25 27 29 31 3310

minus2

10minus1

100

p

C2N

βν

α=0763

(e)

FIG 4 (Color online) In (a) (d) (g) fraction of nodes in the second largest component C2 is plotted as a function of edge density p forα = 07076308 respectively (b) (e) (h) show their rescaling C2N

βν respectively (c) (f) (i) show C2Nβν versus (p minus pprime

c)N1ν indicating

the validity of Eq (7) The system size goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

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CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

α = 06065070750808509 As shown in Fig 1(b)we see C2 (the fractional size of component C2) spikeinstantaneously at the first discontinuous transition then itdisappears and we observe a seemingly continuous percolationtransition when C2 again emerges at some point in thesupercritical regime

To investigate the continuous nature of percolation transi-tion of C2 we make use of finite size scaling [51] If a phasetransition is continuous every variable X is believed to bescale independent and obeys the following finite size scalingform near the percolation threshold pc

X = NminusθνF [(p minus pc)N1ν]

where θ and ν are critical exponents N is the system sizeand F is a universal function If the dimension of the system d

exceeds the upper critical dimension dc the finite-size scalingform for a variable X is assumed to satisfy

X = NminusθνF [(p minus pc)N1ν] (5)

where ν = dcνm and νm is the mean-field value of ν As aresult X follows a power law at p = pc X sim Nminusθν Random

networks display a mean-filed type critical behavior [52]and lack any space dimensionality essentially d = infin Herewe consider two variables in the percolation process thefraction of nodes in the second largest component C2 and thesusceptibility χ The susceptibility χ is defined as the standarddeviation of the size of the second largest component

χ = N

radiclangC2

2rang minus 〈C2〉2 (6)

We assume that C2 and χ obey the following scaling relations

C2 = NminusβνF (1)[(p minus pprimec)N1ν] (7)

χ = NγνF (2)[(p minus pprimec)N1ν] (8)

where F (1) and F (2) are different universal functions (howeverthey are related to each other) pprime

c denotes the percolationthreshold of the second phase transition and βγν are criticalexponents characterizing the transition We can deduce fromthe definition of χ [Eq (6)] and the scaling relations inEqs (7) and (8) at p = pprime

c

βν = 1 minus γ ν (9)

35 4 45 5 550

0002

0004

0006

0008

001

p

χN

α=07(a)

3 35 4 45 50

0002

0004

0006

0008

001

p

χN

α=08(c)

25 3 35 40

0002

0004

0006

0008

001

p

χN

α=0763(b)

104

105

106

107

10minus3

10minus2

N

y=minus0307xminus2277

y=minus0323xminus2602

α=07(d)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0254xminus2606

α=0763

y=minus0251xminus3197

(e)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0237xminus2821

α=08

y=minus0231xminus3416

(f) C2(p

crsquo)

χ(pcrsquo)N

FIG 5 (Color online) In (a) (b) (c) χN is plotted as a function of p for α = 07076308 respectively showing the peak of χN

moves leftward as system size increases The dashed lines are the percolation threshold obtained from numerical simulation through Eq (8)The peaks of χN is moving leftward to the percolation threshold pprime

c which indicates the continuous nature of the second phase transition In(d) (e) (f) C2 and χN at the percolation threshold pprime

c is plotted as a function of system sizes for α = 07076308 respectively The systemsize goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-6

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

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CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

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PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

the following condition holds at t = t(kfinal)

limNrarrinfin

P (ktN ) = α (2)

We measure k(tc)N and kfinalN with system sizes N = 106

for over 100 realizations We find that at the point where thelargest jump of C1 occurs namely tcN for all α isin [06095]k(tc)N = C1 gt 05 averaged over 1000 realizations Dueto this fact once t gt tc the growth of C1 can only occurdue to the mechanism of direct growth [21] in particularthe edge connecting the largest component and anothersmaller component results in the growth of C1 Thereforethe probability of sampling a random edge which leads to thegrowth of C1 is 2(1 minus C1)C1 and thus P (ktN ) satisfies

limNrarrinfin

P (ktN ) = 1 minus 2(1 minus C1)C1 (3)

Combining Eqs (2) and (3) we obtain C1 at t(kfinal) for infinitesystem

C1 = 1 + radic2α minus 1

2 (4)

Figure 3(a) shows the dashed line of Eq (4) intersects k(tc)Nand kfinalN at the same point αc = 0763 within error bars

for system size N = 106 For α lt αc k(tc)N gt 1+radic2αminus12

(blue circles lie above the dashed line) and P (ktcN ) gt αTherefore the stage k stops increasing immediately after thelargest jump of C1 and kfinalN is almost the same as k(tc)N(blue circles and red triangles overlap) However for α gt αck(tc)N lt 1+radic

2αminus12 (blue circles lie below the dashed line)

Therefore both the stage k and C1 increase until Eq (4) holds(red triangles and the dashed line overlap)

IV CONTINUOUS TRANSITION OF THE SECONDLARGEST COMPONENT

The size of the largest component stops increasing at thepoint t(kfinal)N (which is either at the discontinuous phasetransition point or in the supercritical regime dependent onthe value of α) Yet the second largest component continuesgrowing at this point Two natural and interesting questionsfollow Is there a second giant component emerging at somestep t gt t(kfinal) If such percolation transition occurs is ita discontinuous phase transition as we find for the orderparameter C1 To answer these questions we first numericallymeasure C2 as a function of the edge density p equiv tN for

35 37 39 41 43 4510

minus4

10minus3

10minus2

10minus1

p

C2

α=07(a)

32 34 36 38 4 4210

minus4

10minus3

10minus2

10minus1

p

C2

α=08(g)

3 32 34 36 38 410

minus2

10minus1

100

p

C2N

βν

α=08(h)

22 24 26 28 3 3210

minus4

10minus3

10minus2

10minus1

p

C2

α=0763(d)

minus10 minus5 0 5 10006

01

02

03

04

(pminuspcrsquo)N1ν

C2N

βν

α=0763(f)

minus10 minus5 0 5 10002

004

006

01

02

(pminuspcrsquo)N1ν

C2N

βν

α=08(i)

35 37 39 41 43 4510

minus2

10minus1

100

p

C2N

βν

α=07

(b)

minus10 minus5 0 5 10006

008

01

02

03

(pminuspcrsquo)N1ν

C2N

βν

α=07(c)

23 25 27 29 31 3310

minus2

10minus1

100

p

C2N

βν

α=0763

(e)

FIG 4 (Color online) In (a) (d) (g) fraction of nodes in the second largest component C2 is plotted as a function of edge density p forα = 07076308 respectively (b) (e) (h) show their rescaling C2N

βν respectively (c) (f) (i) show C2Nβν versus (p minus pprime

c)N1ν indicating

the validity of Eq (7) The system size goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-5

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

α = 06065070750808509 As shown in Fig 1(b)we see C2 (the fractional size of component C2) spikeinstantaneously at the first discontinuous transition then itdisappears and we observe a seemingly continuous percolationtransition when C2 again emerges at some point in thesupercritical regime

To investigate the continuous nature of percolation transi-tion of C2 we make use of finite size scaling [51] If a phasetransition is continuous every variable X is believed to bescale independent and obeys the following finite size scalingform near the percolation threshold pc

X = NminusθνF [(p minus pc)N1ν]

where θ and ν are critical exponents N is the system sizeand F is a universal function If the dimension of the system d

exceeds the upper critical dimension dc the finite-size scalingform for a variable X is assumed to satisfy

X = NminusθνF [(p minus pc)N1ν] (5)

where ν = dcνm and νm is the mean-field value of ν As aresult X follows a power law at p = pc X sim Nminusθν Random

networks display a mean-filed type critical behavior [52]and lack any space dimensionality essentially d = infin Herewe consider two variables in the percolation process thefraction of nodes in the second largest component C2 and thesusceptibility χ The susceptibility χ is defined as the standarddeviation of the size of the second largest component

χ = N

radiclangC2

2rang minus 〈C2〉2 (6)

We assume that C2 and χ obey the following scaling relations

C2 = NminusβνF (1)[(p minus pprimec)N1ν] (7)

χ = NγνF (2)[(p minus pprimec)N1ν] (8)

where F (1) and F (2) are different universal functions (howeverthey are related to each other) pprime

c denotes the percolationthreshold of the second phase transition and βγν are criticalexponents characterizing the transition We can deduce fromthe definition of χ [Eq (6)] and the scaling relations inEqs (7) and (8) at p = pprime

c

βν = 1 minus γ ν (9)

35 4 45 5 550

0002

0004

0006

0008

001

p

χN

α=07(a)

3 35 4 45 50

0002

0004

0006

0008

001

p

χN

α=08(c)

25 3 35 40

0002

0004

0006

0008

001

p

χN

α=0763(b)

104

105

106

107

10minus3

10minus2

N

y=minus0307xminus2277

y=minus0323xminus2602

α=07(d)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0254xminus2606

α=0763

y=minus0251xminus3197

(e)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0237xminus2821

α=08

y=minus0231xminus3416

(f) C2(p

crsquo)

χ(pcrsquo)N

FIG 5 (Color online) In (a) (b) (c) χN is plotted as a function of p for α = 07076308 respectively showing the peak of χN

moves leftward as system size increases The dashed lines are the percolation threshold obtained from numerical simulation through Eq (8)The peaks of χN is moving leftward to the percolation threshold pprime

c which indicates the continuous nature of the second phase transition In(d) (e) (f) C2 and χN at the percolation threshold pprime

c is plotted as a function of system sizes for α = 07076308 respectively The systemsize goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-6

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

052130-7

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

052130-8

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

α = 06065070750808509 As shown in Fig 1(b)we see C2 (the fractional size of component C2) spikeinstantaneously at the first discontinuous transition then itdisappears and we observe a seemingly continuous percolationtransition when C2 again emerges at some point in thesupercritical regime

To investigate the continuous nature of percolation transi-tion of C2 we make use of finite size scaling [51] If a phasetransition is continuous every variable X is believed to bescale independent and obeys the following finite size scalingform near the percolation threshold pc

X = NminusθνF [(p minus pc)N1ν]

where θ and ν are critical exponents N is the system sizeand F is a universal function If the dimension of the system d

exceeds the upper critical dimension dc the finite-size scalingform for a variable X is assumed to satisfy

X = NminusθνF [(p minus pc)N1ν] (5)

where ν = dcνm and νm is the mean-field value of ν As aresult X follows a power law at p = pc X sim Nminusθν Random

networks display a mean-filed type critical behavior [52]and lack any space dimensionality essentially d = infin Herewe consider two variables in the percolation process thefraction of nodes in the second largest component C2 and thesusceptibility χ The susceptibility χ is defined as the standarddeviation of the size of the second largest component

χ = N

radiclangC2

2rang minus 〈C2〉2 (6)

We assume that C2 and χ obey the following scaling relations

C2 = NminusβνF (1)[(p minus pprimec)N1ν] (7)

χ = NγνF (2)[(p minus pprimec)N1ν] (8)

where F (1) and F (2) are different universal functions (howeverthey are related to each other) pprime

c denotes the percolationthreshold of the second phase transition and βγν are criticalexponents characterizing the transition We can deduce fromthe definition of χ [Eq (6)] and the scaling relations inEqs (7) and (8) at p = pprime

c

βν = 1 minus γ ν (9)

35 4 45 5 550

0002

0004

0006

0008

001

p

χN

α=07(a)

3 35 4 45 50

0002

0004

0006

0008

001

p

χN

α=08(c)

25 3 35 40

0002

0004

0006

0008

001

p

χN

α=0763(b)

104

105

106

107

10minus3

10minus2

N

y=minus0307xminus2277

y=minus0323xminus2602

α=07(d)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0254xminus2606

α=0763

y=minus0251xminus3197

(e)C

2(p

crsquo)

χ(pcrsquo)N

104

105

106

107

10minus3

10minus2

N

y=minus0237xminus2821

α=08

y=minus0231xminus3416

(f) C2(p

crsquo)

χ(pcrsquo)N

FIG 5 (Color online) In (a) (b) (c) χN is plotted as a function of p for α = 07076308 respectively showing the peak of χN

moves leftward as system size increases The dashed lines are the percolation threshold obtained from numerical simulation through Eq (8)The peaks of χN is moving leftward to the percolation threshold pprime

c which indicates the continuous nature of the second phase transition In(d) (e) (f) C2 and χN at the percolation threshold pprime

c is plotted as a function of system sizes for α = 07076308 respectively The systemsize goes from N = 20 000 (the dashed magenta line) to 1 620 000 (the solid red line) via successive doubling

052130-6

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

052130-7

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

052130-8

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2479

α=07(a)

100

101

102

10310

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2504

α=0763(b)

100

101

10210

minus6

10minus5

10minus4

10minus3

10minus2

10minus1

s

n(s)

τ=minus2486

α=08(c)

FIG 6 (Color online) Component size distribution n(s) at the percolation threshold pprimec for α = 07 (a) 0763 (b) 08 (c) respectively

Since Eqs (7) and (8) imply that the curves of NβνC2 vsp and Nminusγ νχ vs p for different systems sizes cross at asingle point at p = pprime

c both the relative size of the secondlargest component C2 and the susceptibility χ can be used forthe determination of percolation threshold pprime

c More preciselywe implement the minimization technique in Ref [53] Foreach system size N we use Monte Carlo data for C2 vsp and χN vs p to create functions gN = NβνC2 andlN = Nminusγ νχ respectively The value of p that minimizes theobjective function P = sum

NiltNj (gNi minus gNj )2 is the estimationof percolation threshold pprime

c Similarly the value of p whichminimizes the objective function Q = sum

NiltNj (lNi minus lNj )2 isan alternative way to estimate pprime

c In our following numericalsimulations we find a perfect agreement within error barsbetween the two different methods

In Figs 4(a) 4(d) and 4(g) we plot the order pa-rameter C2 as a function of edge density p for α =07076308 in the BFW model respectively Five curvesin each figure stand for Monte Carlo data averaged over1000 realizations for five different system sizes namelyN = 20 00060 000180 000540 0001 620 000 To identifythe percolation threshold pprime

c of C2 and two scaling exponentsβνγ ν we first implement the minimization technique in theobjective function P Figures 4(b) 4(e) and 4(h) show that allcurves of C2N

βν vs p cross at a single percolation transitionpoint pprime

c = 3865 plusmn 00053555 plusmn 00052725 plusmn 0003 forα = 07076308 respectively The corresponding exponentratios βν are 031 plusmn 003025 plusmn 004024 plusmn 004 for α =07076308 respectively In Figs 4(c) 4(f) and 4(i) thedata collapses show the profiles of the universal scalingfunction F (1) of Eq (7) where we find the exponent 1ν =036 plusmn 001036 plusmn 001037 plusmn 001 for α = 07076308respectively

In Figs 5(a) 5(b) and 5(c) χN is plotted as a function ofp where the dashed lines show the location of the percolationthreshold through minimization of the objective functionQ We find that percolation threshold obtained throughminimization technique in the objective functions P and Q

show perfect agreement within error bars Figures 5(d)ndash5(f)display C2 and χN at p = pprime

c as a function of system size N

for α = 07076308 respectively The slope of dashed lineswhich are the best fit of points represent the exponent ratiosminusβν and γ ν minus 1 in scaling relations Eqs (7) and (8) Weobtain that 1 minus γ ν = 032 plusmn 003025 plusmn 004023 plusmn 005for α = 07076308 respectively which agree with thecorresponding exponent ratios βν Therefore the validity ofEq (9) has been verified The exponent ratios βνγ ν forα = 07 are the same within error bars as observed for ER [17]suggesting they belong to the same universality class Themeasurement of exponents βγ and ν verify the fact that thepercolation transition is continuous

Next we numerically measure at the percolation thresholdpprime

c the number of components of size s divided by the numberof nodes N denoted as n(s) Figures 6(a)ndash6(c) show thatn(s) follow a power law distribution for α = 07076308with respective scaling exponent τ = 248 plusmn 001250 plusmn001249 plusmn 001 which is further evidence for a continuousphase transition The continuous phase transition of the secondlargest component is a consequence of a relatively large valueof the cap k throughout the continuous growth process ofC2 This stands in contrast to the subcritical regime for C1

where small values of k keep components similar in size bythe mechanism of growth by overtaking

The ratios of the scaling exponents βνγ ν and the perco-lation threshold pprime

c are measured by minimization techniquewith results shown in Table II For α isin [06095] we find

TABLE II Summary of numerical results obtained from minimization technique

α 06 065 068 07 072 075 0763(αc) 08 085 09

βν 026(4) 031(4) 030(3) 031(3) 027(3) 030(3) 025(4) 024(5) 028(4) 022(5)γ ν 074(4) 068(4) 068(1) 068(3) 073(3) 068(3) 075(4) 077(5) 072(4) 078(5)τ 250 (1) 248(2) 249(2) 248(1) 249(1) 249(2) 250(1) 249(1) 246(2) 248(2)pprime

c 7595(6) 5320(5) 4385(5) 3865(5) 3455(4) 2860(4) 2725(3) 3555(5) 5215(5) 8647(7)

052130-7

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

052130-8

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

CHEN NAGLER CHENG JIN SHEN ZHENG AND DrsquoSOUZA PHYSICAL REVIEW E 87 052130 (2013)

06 065 07 075 08 085 09086

088

09

092

094

096

α

C1

(a)

06 065 07 075 08 085 09004

006

008

01

012

014

α

C2

(b)

06 065 07 075 08 085 092

4

6

8

α

p crsquo

(c)

FIG 7 (Color online) (a) The asymptotic fraction of nodes in thelargest component is plotted as a function of α (b) The asymptoticfraction of nodes in the second largest component is plotted as afunction of α (c) The percolation threshold pprime

c of the second phasetransition is plotted as a function of α The minimal values of C1 andpprime

c the maximal value of C2 are obtained at the same critical pointαc = 0763 plusmn 0001 marking also the point with the minimal delaybetween the emergence of C1 and C2

strong numerical evidence that the percolation transition isindeed continuous We also deduce from Table II that theminimal percolation threshold of the order parameter C2 isobtained for αc = 0763 plusmn 0002 as also shown explicitly inFig 7(c) This observation shows perfect agreement with thelocation where t(kfinal) and kfinalN attain their minimums[Fig 3(a)] The underlying mechanism which accounts forthe minimal percolation threshold obtained at αc = 0763 plusmn0002 is actually related with the behavior of C1 (or stage k) insupercritical regime Since if α lt αc both k(tc)N and tcN

decrease with α from numerical results obtained in Sec III theemergence of the second giant component is earlier for largerα However if α gt αc C1 keeps on augmenting after tcN anda finite fraction of edges is added before t(kfinal)N Thereforethe competition of edges enhancing the largest component andthe second largest component delays the onset of the secondgiant component In particular the emergence of the secondgiant component is more delayed for larger α if α gt αc Thus

we observe the minimal percolation threshold of the orderparameter C2 is derived for αc = 0763 plusmn 0002

V SUMMARY AND DISCUSSION

In this paper we have analyzed the BFW model inthe regime α isin [06095] where only one giant componentappears in a discontinuous phase transition The growthcessation of the largest component in the supercritical regimeresults in the emergence of a second giant component at somedelayed time We have carried out finite-size scaling to studythe critical behavior of the second phase transition and theresults establish that the transition is continuous We have alsoestablished that there exists an interesting inflection point atαc = 0763 plusmn 0002 that is related to many properties of thesystem As derived in Eq (4) for α lt αc C1 stops growingthe instant it emerges whereas for α gt αc C1 grows in thesupercritical regime As shown in Fig 7 at αc the asymptoticsize of C1 is minimized that of C2 is maximized and the delaybetween the emergence of C1 and C2 is minimized

There is much recent interest in understanding the forma-tion mechanism of multiple giant components in percolation[38405455] In Refs [3840] the modified BFW model withα lt 052 exhibits multiple giant components which emergein a discontinuous transition The underlying mechanismsresponsible for the formation of multiple giant componentsare (i) the domination of overtaking processes in the growth ofC1 which result in the coexistence of many large components inthe critical window and (ii) the dominance of the sampling ofintracomponent edges in the supercritical region which avoidsmerging the giant components In Ref [54] the so calledmulti-ER model shows multiple giant components appearin a continuous transition The formation of multiple giantcomponents is attributed to the relatively low probability formerger of large components in the critical window In thispaper the BFW model with α isin [06095] provides a novelformation mechanism resulting in multiple giant componentswhich is the growth cessation of C1 in supercritical regimeresulting in the emergence of a second giant component

The hybrid of continuous and discontinuous phase transi-tions observed in the modified BFW model studied here isan unusual characteristic in classical statistical mechanics ofcritical phenomena In a recent study a competitive percolationmodel called the Nagler-Tiessen-Gutch model (NTG) [50] hasbeen proposed in which three nodes are randomly sampled ateach step and an edge is added between two nodes that reside incomponents most similar in size The NTG model undergoesboth continuous and discontinuous phase transitions as wellHowever there is a different main feature between the NTGmodel and the modified BFW model with α isin [0609] Inthe NTG model the double giant components would mergeinfinitely many times in the supercritical regime leading toinfinite discontinuous transitions of C1 and infinite continuoustransitions of C2 while in the modified BFW model thereare asymptotically stable double giant components that nevermerge

It is a longstanding challenge to substantiate the dis-continuity of phase transition Here we have investigatedthe discontinuity of phase transition of the BFW model byestimating the asymptotic values of C Pinfin and M prime

2 through

052130-8

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9

PHASE TRANSITIONS IN SUPERCRITICAL EXPLOSIVE PHYSICAL REVIEW E 87 052130 (2013)

numerical simulations It would be interesting to understandthe dependence of C Pinfin and M prime

2 on α in the function of g(k)in a rigorous analytical manner which will be our future study

ACKNOWLEDGMENTS

We are grateful to Robert Ziff for useful commentsThis work was supported by the Defense Threat Reduction

Agency Basic Research Award No HDTRA1-10-1-0088the Army Research Laboratory under Cooperative Agree-ment No W911NF-09-2-0053 the 973 National Basic Re-search Program of China (Grant No 2005CB321902) andthe 973 National Basic Research Program of China (No2013CB329602) the National Natural Science Foundation ofChina under Grant Nos 61232010 and 61202215

[1] S H Strogatz Nature 410 268 (2001)[2] M E J Newman D J Watts and S H Strogatz Proc Natl

Acad Sci 99 2566 (2002)[3] C Song S Havlin and H A Makse Nat Phys 2 275 (2006)[4] S V Buldyrev R Parshani G Paul H E Stanley and S Havlin

Nature 464 1025 (2010)[5] J Kim P L Krapivsky B Kahng and S Redner Phys Rev E

66 055101 (2002)[6] H D Rozenfeld L K Gallos and H A Makse Eur Phys J

B 75 305 (2010)[7] Cristopher Moore and M E J Newman Phys Rev E 61 5678

(2000)[8] M A Serrano and M Boguna Phys Rev Lett 97 088701

(2006)[9] S N Dorogovtsev A V Goltsev and J F F Mendes Rev Mod

Phys 80 1275 (2008)[10] R Parshani S Carmi and S Havlin Phys Rev Lett 104

258701 (2010)[11] M Ausloos and R Lambiotte Physica A 382 1621 (2007)[12] Carlos P Roca Moez Draief and Dirk Helbing

arXiv11010775[13] P Erdos and A Renyi Publ Math Inst Hung Acad Sci 5 17

(1960)[14] D Achlioptas R M DrsquoSouza and J Spencer Science 323

1453 (2009)[15] Y S Cho J S Kim J Park B Kahng and D Kim Phys Rev

Lett 103 135702 (2009)[16] F Radicchi and S Fortunato Phys Rev Lett 103 168701

(2009)[17] F Radicchi and S Fortunato Phys Rev E 81 036110 (2010)[18] R M Ziff Phys Rev Lett 103 045701 (2009)[19] R M Ziff Phys Rev E 82 051105 (2010)[20] R A da Costa S N Dorogovtsev A V Goltsev and J F F

Mendes Phys Rev Lett 105 255701 (2010)[21] J Nagler A Levina and M Timme Nat Phys 7 265 (2011)[22] O Riordan and L Warnke Science 333 322 (2011)[23] P Grassberger C Christensen G Bizhani S-W Son and

M Paczuski Phys Rev Lett 106 225701 (2011)[24] H K Lee B J Kim and H Park Phys Rev E 84 020101(R)

(2011)[25] Y S Cho B Kahng and D Kim Phys Rev E 81 030103(R)

(2010)[26] N A M Araujo and H J Herrmann Phys Rev Lett 105

035701 (2010)[27] A A Moreira E A Oliveira S D S Reis H J Herrmann

and J S Andrade Phys Rev E 81 040101(R) (2010)

[28] K J Schrenk N A M Araujo and H J Herrmann Phys RevE 84 041136 (2011)

[29] W Choi S-H Yook and Y Kim Phys Rev E 84 020102(R)(2011)

[30] Y S Cho and B Kahng Phys Rev E 84 050102(R) (2011)[31] Y S Cho Y W Kim and B Kahng J Stat Mech (2012)

P10004[32] K Panagiotou R Spohel A Steger and H Thomas Elec Notes

in Discrete Math 38 699 (2011)[33] S Boettcher V Singh and R M Ziff Nature Communications

3 787 (2012)[34] G Bizhani M Paczuski and P Grassberger Phys Rev E 86

011128 (2012)[35] L Cao and J M Schwarz Phys Rev E 86 061131 (2012)[36] Y S Cho and B Kahng Phys Rev Lett 107 275703 (2011)[37] K J Schrenk A Felder S Deflorin N A M Araujo R M

DrsquoSouza and H J Herrmann Phys Rev E 85 031103 (2012)[38] W Chen and R M DrsquoSouza Phys Rev Lett 106 115701

(2011)[39] T Bohman A Frieze and N C Wormald Random Structures

amp Algorithms 25 432 (2004)[40] W Chen Z Zheng and R M DrsquoSouza Europhys Lett 100

66006 (2012)[41] M Bengtsson and S Kock J Bus Ind Mark 14 178 (1999)[42] S A Frank Evolution Int J Org Evol 57 693 (2003)[43] J Hirshleifer Am Econ Rev 68 238 (1978)[44] J Spencer Not Am Math Soc 57 720 (2010)[45] E Ben-Naim and P L Krapivsky J Phys A 38 L417 (2005)[46] A Asztalos and Z Toroczkai Europhys Lett 92 50008 (2010)[47] R M Anderson and R M May Infectious Diseases in Humans

(Oxford University Press Oxford 1992)[48] S S Manna and A Chatterjee Physica A 390 177 (2011)[49] O Riordan and L Warnke Phys Rev E 86 011129 (2012)[50] J Nagler T Tiessen and H W Gutch Phys Rev X 2 031009

(2012)[51] D P Landau and K Binder A Guide to Monte Carlo Simulations

in Statistical Physics (Cambridge University Press CambridgeEngland 2000)

[52] H Hong M Ha and H Park Phys Rev Lett 98 258701(2007)

[53] N Bastas K Kosmidis and P Argyrakis Phys Rev E 84066112 (2011)

[54] Y Zhang W Wei B Guo R Zhang and Z Zheng Phys RevE 86 051103 (2012)

[55] R Zhang W Wei B Guo Y Zhang and Z Zheng Physica A392 1232 (2013)

052130-9