PHYSICAL REVIEW E 85, 031103 (2012)

Bohman-Frieze-Wormald model on the lattice, yielding a discontinuous percolation transition

K. J. Schrenk,1,* A. Felder,1,† S. Deflorin,1,‡ N. A. M. Araujo,1,§ R. M. D’Souza,2,3,‖ and H. J. Herrmann1,4,¶1Computational Physics for Engineering Materials, IfB, ETH Zurich, Schafmattstrasse 6, CH-8093 Zurich, Switzerland

2University of California, Davis, California 95616, USA3Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

4Departamento de Fısica, Universidade Federal do Ceara, Campus do Pici, 60451-970 Fortaleza, Ceara, Brazil(Received 11 November 2011; revised manuscript received 28 January 2012; published 2 March 2012)

The BFW model introduced by Bohman, Frieze, and Wormald [Random Struct. Algorithms 25, 432 (2004)],and recently investigated in the framework of discontinuous percolation by Chen and D’Souza [Phys. Rev. Lett.106, 115701 (2011)], is studied on the square and simple-cubic lattices. In two and three dimensions, we findnumerical evidence for a strongly discontinuous transition. In two dimensions, the clusters at the threshold arecompact with a fractal surface of fractal dimension df = 1.49 ± 0.02. On the simple-cubic lattice, distinct jumpsin the size of the largest cluster are observed. We proceed to analyze the tree-like version of the model, where onlymerging bonds are sampled, for dimension two to seven. The transition is again discontinuous in any considereddimension. Finally, the dependence of the cluster-size distribution at the threshold on the spatial dimension isalso investigated.

DOI: 10.1103/PhysRevE.85.031103 PACS number(s): 64.60.ah, 64.60.al, 89.75.Da

I. INTRODUCTION

Percolation is a classical model in statistical physics that,despite consisting of simple geometrical rules, has foundapplication in a broad range of problems [1,2]. In classical(random) percolation, a fraction p of sites or bonds areoccupied uniformly at random. At a critical occupation fractionpc, the system undergoes a continuous transition from anonpercolating to a percolating state. A recent work byAchlioptas, D’Souza, and Spencer [3] hinted at the possibilityof a discontinuous percolation transition by slightly modifyingthe bond selection rules of evolving graphs, coining the termexplosive percolation. This surprising result led to intensiveresearch efforts studying the model on different topologies anddimensions, yielding ambiguous results regarding the natureof the transition [4–18]. More recently, the transition in theoriginal model was formally demonstrated to be continuousin the mean-field limit [19] and there is evidence that thesame applies to other models [20–23]. Alternative models havebeen devised and analyzed in detail [24–33], several of themwith clear signs supporting the hypothesis of a discontinuoustransition.

In particular, Chen and D’Souza considered the percolationproperties on the random graph of the BFW model [34,35],originally introduced by Bohman, Frieze, and Wormald [36],reporting a discontinuous transition with several giant compo-nents in the thermodynamic limit. In general, the behavior of asystem undergoing a transition depends on the dimension [37].In this paper, we analyze the original BFW model on the 2Dsquare lattice and on the 3D cubic lattice, as well as the tree-likeversion up to dimension seven.

*jschrenk@ethz.ch†afelder@ethz.ch‡sdeflori@ethz.ch§nuno@ethz.ch‖raissa@cse.ucdavis.edu¶hans@ifb.baug.ethz.ch

The manuscript is organized as follows. In Sec. II we discussthe BFW model, adapted to the square lattice, along withsome definitions. Section III contains the extension to threedimensions. The tree-like version is analyzed in Sec. IV. Weconclude with the final remarks in Sec. V.

II. BFW MODEL ON THE SQUARE LATTICE

The BFW model has recently been analyzed by Chenand D’Souza [34,35] on the random graph, by extendingthe algorithm initially introduced by Bohman, Frieze, andWormald [36]. Here, we consider this model on a square latticeof linear size L with periodic boundary conditions and numberof sites N , where N = L2. Initially, all sites are occupiedand all bonds are unoccupied (empty), such that there are N

clusters of unitary size. As in Refs. [34,35], one occupies thebonds according to the following procedure. Let u be the totalnumber of selected bonds, t the number of occupied bonds,and k the stage of the process, initially set to k = 2. The firstbond is occupied at random, such that t = u = 1. Then, at eachstep u,

(1) One bond, among the unoccupied ones, is selecteduniformly at random.

(2) l is measured as the maximum cluster size if the selectedbond is occupied.

(3) If l � k, go to (4). Else, if t/u � g(k) = 1/2 +√1/(2k), go to (5). Else, increment k by one and go to (3).(4) Occupy the selected bond.(5) Increment u by one.Note that the above procedure, in (1), only samples

unoccupied bonds, while in Refs. [34–36] edges are chosenuniformly at random from all edges, whether occupied ornot. In adapting random graph models to the lattice, wesolely sample unoccupied bonds since in classical (random)percolation the control parameter is the fraction of occupiedbonds [4,5,7,8,21]. In contrast, for the BFW model on therandom graph, by sampling all bonds, since sufficiently manynew links can be accepted asymptotically, more than onestable macroscopic cluster is obtained [34,35]. The procedure

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K. J. SCHRENK et al. PHYSICAL REVIEW E 85, 031103 (2012)

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FIG. 1. (Color online) Fraction of sites in the largest cluster P∞(solid red line), stage per site k/N (◦), occupied bonds per step t/u

(dashed upper blue line), and g(k) (dotted lower black line) as afunction of the fraction of occupied bonds p, for the BFW modelon the square lattice. As reported in Refs. [34,36] for the randomgraph, k is essentially equal to smax. Therefore, the correspondingcurves in the figure, P∞ and k/N , almost overlap. The upper insetshows a close-up view of the upper-left corner of the main plot, asindicated by the box. The system size is N = 5122, and the measuredvalues are averages of over more than 105 samples. In the lower insetwe see the same variables as the main plot, measured for a singlerealization.

is applied iteratively while the system evolves from t = 0,corresponding to a bond occupation fraction of p = t/(2N ) =0, to t = 2N , where all bonds in the system are occupied,i. e., p = 1. By construction, at each stage, the size of thelargest cluster can be at most smax � k (see Fig. 1) [36]. Toillustrate the evolution of this process, we monitor in Fig. 1 theusual order parameter for the percolation transition, defined asthe fraction of sites in the largest cluster P∞ = smax/N . Oneobserves that the transition is delayed, compared to classicalpercolation where pc = 1/2, and the growth of the orderparameter at the threshold appears to be more pronounced thanin the classical case (numerical evidence discussed below). Inaddition, Fig. 1 shows the behavior of the stage per site k/N ,which grows as the system evolves and which imposes a boundon the order parameter, P∞ � k/N . The fraction of acceptedbonds t/u and the value of g(k) are also shown as functionsof the fraction of occupied bonds p. These curves illustratethe mechanism of the BFW model: if the fraction of acceptedbonds were to drop below g(k), the stage k is incremented,which decreases g(k) (see insets of Fig. 1). Starting fromfull acceptance, t/u decreases, approaching the asymptoticvalue of 1/2. The mechanism promotes the homogenization ofcluster sizes and delays the transition. Once percolation occurs,it is more abrupt than in the case of uniformly random bondoccupation. When k/N = P∞ = 1, with p < 1, the entireset of unoccupied bonds consists of redundant bonds, thatis unoccupied bonds internal to existing clusters, which canbe occupied without affecting the size of clusters. In Fig. 2we see the fraction of sites in the largest cluster P∞ as afunction of the fraction of occupied bonds p for differentlattice sizes L. Typical snapshots of the evolving system areshown in Fig. 3, for fractions of occupied bonds p = 0.5, 0.9,and 0.95.

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FIG. 2. (Color online) Fraction of sites in the largest cluster P∞as a function of the fraction of occupied bonds p on the square latticefor different lattice sizes L (from left to right: 32, 64, 128, 256, 512,1024, 2048, 4096, and 8192).

To quantify this observation, we make the following con-siderations. If the largest change J in the order parameter P∞is finite in the thermodynamic limit, a percolation transitionis called discontinuous, as opposed to continuous transitionswhere the jump J vanishes for L → ∞ [38]. Additionalways to distinguish between continuous and discontinuoustransitions are to consider the behavior of the standarddeviation of the order parameter and of

M ′2 = M2 − s2

max/N, (1)

where M2 = ∑i s

2i /N is the second moment of the cluster-

size distribution and si is the size of cluster i. Figure 4shows the size independence of the largest change 〈J 〉j andthe order parameter 〈P∞〉j immediately before the biggestjump, where the average 〈·〉j means averaging the variableswhen the biggest jump occurs for each realization, and themaximum of M ′

2. One observes that all three quantities arewithin error bars asymptotically independent on the systemsize: J = 0.415 ± 0.005, P∞ = 0.585 ± 0.005, and M ′

2 =0.184 ± 0.006. These results suggest that the transition isdiscontinuous [24,32,39], as was previously reported for therandom graph [34,35]. We remark that the numerical valuesof J , P∞, and M ′

2 are within error bars identical to theones found for a different discontinuous percolation modelin Refs. [24,32].

Unless indicated otherwise, on the square lattice, resultshave been averaged over 108 samples for the smallest systemsize (322) and 70 samples for the largest one (81922), anderror bars are smaller than the symbol size. Random numbershave been generated with the algorithm proposed in Ref. [40].To keep track of the cluster properties as the fraction ofoccupied bonds increases, we have considered the labelingscheme proposed by Newman and Ziff [41,42], related to theHoshen-Kopelman algorithm [43].

To determine the bond occupation fraction pc

at which the percolation transition occurs, wemeasure for every sample the position pc,J ofthe largest jump J , the position pc,M of the maximumof M ′

2, and the point pc,S at which spanning in one specificdirection occurs. Extrapolating these estimators to thethermodynamic limit, we find pc = 1.000 ± 0.002. In Fig. 5

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FIG. 3. (Color online) Snapshots of the system evolving accord-ing to the BFW model, obtained on a square lattice with 5122 sites.The following fractions of occupied bonds are shown: (a) p = 0.5,(b) 0.9, and (c) 0.95. We note that in the thermodynamic limit pc

seems to be equal to unity (see Fig. 10).

we see pc − pc,X as a function of the linear system size L, forall three estimators. Asymptotically, the points follow a powerlaw (pc − pc,X) ∼ L−a with a = 0.5 ± 0.2. In addition, withthese results we can obtain a scaling plot for the standard

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FIG. 4. Order parameter P∞ (∗) immediately before the jump,jump size J (�), and maximum of the second moment of the cluster-size distribution M ′

2 (◦) as a function of the linear system size L.All three quantities are asymptotically independent of the systemsize. For large system sizes, we estimate the following values: J =0.415 ± 0.005, P∞ = 0.585 ± 0.005, and M ′

2 = 0.184 ± 0.006. Thesolid lines represent the estimated mean value for each quantity. Theseresults are clear signs for a discontinuous transition [24,32,39], aspreviously found for the random graph [34,35].

deviation of the order parameter χ∞,

χ∞ =√⟨

P 2∞⟩ − 〈P∞〉2 , (2)

where 〈·〉 is an average over independent realizations. A datacollapse is obtained if χ∞(p,L) is plotted as a function of(p − pc)La; see Fig. 6. The fact that the maximum of χ∞is independent on the system size is further evidence fora discontinuous transition [24,32,39]. Here, χ∞ is definedas an intensive variable [see also Eq. (2)], as the standarddeviation of the largest cluster size per lattice site. In thethermodynamic limit, this quantity is zero for continuoustransitions and nonzero for discontinuous transitions. For theconsidered system sizes, the peak height is independent of L.Results have been averaged over at least 1.5 × 103 samples.

In Fig. 3 one observes that slightly below the percolationthreshold the clusters of the BFW model are compact but witha corrugated surface. This behavior has also been reported

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FIG. 5. pc − pc,X , with pc = 1.000 ± 0.002, as a function ofthe linear system size L for the three estimators, the position ofthe jump pc,J (◦), the position pc,M (∗) of the maximum of M ′

2,and the point pc,S (�) at which spanning in one specific directionoccurs. Asymptotically, we observe (pc − pc,X) ∼ L−a with a =0.5 ± 0.2.

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FIG. 6. (Color online) Scaling plot for the standard deviation ofthe order parameter χ∞ plotted as a function of the scaling variable(p − pc)La , with pc = 1 and a = 0.5, for different linear systemsizes L [32 (�), 64 (×), 128 (∗), 256 (�), 512 (�), 1024 (◦),2048 (•), 4096 (�), and 8192 (�)]. Note that this value of theexponent a agrees within error bars with the one obtained fromthe asymptotic behavior of (pc − pc,X) ∼ L−a , a = 0.5 ± 0.2; seeFig. 5. One observes that χ∞ is nonzero in the thermodynamic limit,as expected for a discontinuous transition.

for another discontinuous percolation model, where the fractaldimension of the cluster perimeter was found to be consistentwith the one of the bridge line in bridge percolation andthe watershed [24,32,44–53]. Compact clusters with fractalsurface have also been found in irreversible aggregation athigh concentration [54]. For the BFW model, we measure atthe stage where the number of clusters in the system equalstwo, for every sample the number A of bonds that wouldconnect these two clusters if they would be occupied. Theinterface size [55] is observed to scale asymptotically with thesystem size, A ∼ Ldf (see Fig. 7). To estimate the fractaldimension more accurately, we plot the local slopes dL =log(AL/AL/2)/ log(2) as a function of the inverse system sizeL−0.5 (see lower right inset of Fig. 7), which yields an exponentof 1.49 ± 0.02. This result is consistent with the fractal dimen-sion of the interface as obtained from box counting (see upperleft inset of Fig. 7), which yields an exponent of 1.49 ± 0.08.Interestingly, this value is different from the one of Ref. [24],where df = 1.23 ± 0.03, but similar to the surface fractaldimension in irreversible aggregation at concentration unity,where df = 1.60 ± 0.08 [54].

We note that for the BFW model, clusters are compactand only the perimeter is fractal, with dimension df . For 2Drandom percolation, both the cluster mass and perimeter arefractals with dimension 91/48 = 1.89583 . . . [1] and df =7/4 = 1.75 [55,56], respectively.

We also simulated the BFW model on the 2D triangularlattice and measured, in analogy to Fig. 4, the size of the jumpJ , the value of the order parameter P∞ immediately before thejump, and the maximum of M ′

2 (not shown). Like on the squarelattice, we found the three quantities to be asymptoticallyindependent on the lattice size: J = 0.415 ± 0.005, P∞ =0.585 ± 0.005, and M ′

2 = 0.183 ± 0.007, within error barsidentical to the results for the square lattice. This is an evidencethat the BFW percolation transition is discontinuous also onthe triangular lattice.

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FIG. 7. Size dependence of the number of interface bonds A (◦).For large systems the interface size scales with the system size, A ∼Ldf . The straight line is a guide to the eye given by 0.824L1.47. In thelower right inset we see the local slopes dL = log(AL/AL/2)/ log(2)(•) as function of the inverse system size L−0.5. Extrapolating toL → ∞ yields an exponent of 1.49 ± 0.02. Box counting resultsare shown in the upper left inset: we see the rescaled number ofoccupied boxes BL−df as a function of the linear box size n for twodifferent system sizes: L = 8192 (�) and L = 4096 (�), the exponentis 1.49 ± 0.08. Results have been averaged over 108 samples for thesmallest system size (162) and 103 samples for the largest one (81922).

III. BEHAVIOR IN THREE DIMENSIONSAND MULTIPLE JUMPS

In Sec. II we considered the BFW model on the squarelattice and found evidence for a discontinuous transition. Weproceed to analyze the BFW model on the simple-cubic lattice.In Fig. 8 we see the analog of Fig. 1 for the simple-cubic lattice,namely, the fraction of sites in the largest cluster P∞, stage persite k/N , occupied bonds per step t/u, and g(k) as a functionof the fraction of occupied bonds p. One observes that thefraction of sites in the largest cluster P∞ grows stepwise (see

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FIG. 8. (Color online) Fraction of sites in the largest cluster P∞(solid red line), stage per site k/N (◦), occupied bonds per step t/u

(dashed upper blue line), and g(k) (dotted lower black line) as afunction of the fraction of occupied bonds p, for the BFW model onthe simple-cubic lattice. The system size is N = 643, the measuredvalues are averages over 103 samples. In the inset we see the samevariables, measured for a single realization.

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FIG. 9. (Color online) Fraction of sites in the largest cluster P∞as a function of the fraction of occupied bonds p on the simple-cubiclattice for different lattice sizes L (from left to right: 8, 16, 32, 64,128, 256, and 512).

also Fig. 9), which hints at the presence of multiple stablecomponents, as reported for the random graph [34,35]. As onthe square lattice (see Fig. 1), the size of the largest clusteris bounded by the stage k. The fraction of accepted bondst/u decreases initially, ultimately reaching the asymptoticvalue of g(k) = 1/2 before increasing slightly once only onecluster remains. In stages where smax is essentially constant,the fraction of accepted bonds increases due to the addition ofredundant bonds.

To analyze this effect in detail, we proceed as follows.For every sample we measure the NJ largest increases inthe order parameter P∞, denoted as Ji , and the fraction ofoccupied bonds pc,J,i at which they occur, and average overmany samples. This definition contains the above ones of J

and pc,J , since J = J1 and pc,J = pc,J,1. Let us consider thesix largest increases in the order parameter; i. e., NJ = 6. Inthe 2D square lattice, only one jump occurs: one observesthat, extrapolated to L → ∞, the occupation fractions pc,J,i

are identical within their error bars [see Fig. 10(a)], and thecorresponding jump sizes Ji are macroscopic (not shown). Thismeans that in the thermodynamic limit the fraction of bondsthat is added between two macroscopic increases of the orderparameter is zero. In this sense, for two dimensions, all jumpshappen at the same instant p = pc. In contrast, as shown inFig. 10(b), in three dimensions, the macroscopic jumps occurat different bond occupation fractions pc,J,i . This is consistentwith multiple stable clusters emerging and coexisting whenadded bonds are mainly redundant ones. In Fig. 11 we seethe jump size J as a function of the fraction of occupiedbonds p for the simple-cubic lattice. We divide the consideredrange of bond occupation fractions p into 512 bins of equalsize. In each bin the maximum of the changes in the orderparameter is measured and averaged over several samples. Inthe plot, one sees four peaks with heights increasing or beingconstant with increasing lattice size. Since these peaks arewell-separated and do not seem to approach each other forincreasing lattice sizes, this corresponds to four macroscopicjumps occurring at distinct values of p. This is consistentwith the results shown in Figs. 10(b) and 8: the arrows inFig. 11 indicate the positions of the four largest increases inthe largest cluster size, as obtained in Fig. 10(b). We remarkthat the number of jumps measured at distinct fractions of

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FIG. 10. Bond occupation fractions pc,J,i at which the six largestjumps in the order parameter occur for the BFW model as a functionof the inverse system size, for (a) the square lattice and (b) the simple-cubic lattice. While for two dimensions all pc,J,i can be extrapolated tothe same value pc = 1.000 ± 0.002, in three dimensions four distinctjumps are visible within error bars for L → ∞ (compare also Fig. 8).For the square lattice, a = 0.5 ± 0.2 and the solid line is a guide to theeye. On the simple-cubic lattice we find pc,1 = 0.756 ± 0.005, pc,2 =0.623 ± 0.007, pc,3 = 0.578 ± 0.006, and pc,4 = 0.539 ± 0.008. For(b) results have been averaged over 108 samples for the smallestsystem size (83) and 40 samples for the largest one (5123).

occupied bonds depends on the resolution, which in Fig. 11is related to the bin size. An interesting possibility would bethat the fraction of sites in the largest cluster P∞ as functionof the fraction of occupied bonds p actually behaves like aDevil’s Staircase, with an infinite hierarchy of jumps. In thenext section, we study the tree-like version of the model wheresolely merging bonds are occupied, i. e., bonds connectingedges belonging to different clusters.

IV. TREE-LIKE BFW MODEL

The BFW model as defined in Sec. II seems to exhibita discontinuous transition on the square and simple-cubiclattices. We found that in three dimensions multiple jumps ofthe order parameter occur at distinct bond occupation fractionsin the thermodynamic limit, which can only occur whenredundant bonds are occupied. We now consider a modifiedversion of the BFW model where redundant bonds are neveroccupied, i. e., bonds connecting sites that are part of thesame cluster are not sampled and, therefore, all clusters aretree-like. This version was also considered for random graphsin Ref. [34] under the name restricted BFW model.

Figure 12 shows P∞ as a function of the fraction of occupiedbonds p for the tree-like BFW model and, for the sake of

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FIG. 11. (Color online) Maximum change J in the order param-eter in the BFW model on the simple-cubic lattice as a function ofthe fraction of occupied bonds p for different linear system sizesL [128 (�), 256 (◦), and 512 (�): peaks become narrower withincreasing L]. The arrows indicate the positions of the four largestjumps, extrapolated to L → ∞: p = 0.756, 0.623, 0.578, and 0.539(see Fig. 10). The range between zero and unity for p was dividedinto 512 bins of equal size and in every bin the maximum of theoccurred changes in the order parameter was measured. Results havebeen averaged over 1.6 × 105 samples for the smallest system size(1283) and 2.4 × 103 samples for the largest one (5123).

comparison, we also include the results for tree-like classicalpercolation. We observe that under this constraint, for theBFW model the transition seems abrupt in all considereddimensions, whereas in the classical case the transition iscontinuous and the percolation thresholds are consistent withthe values expected for the number of clusters at the criticalpoint (see Refs. [57–59] and references therein). In particular,at pc, the number of clusters per site (Nc/N ) in tree-likepercolation is Nc/N = 1 − pcd, which on the square lattice,yields pc ≈ (7 − 3

√3)/4 [57–59]. The insets in Fig. 12 show

the fraction of sites s/N in the three largest clusters asa function of the fraction of occupied bonds p for singlerealizations of the tree-like BFW model [Fig. 12(a)] andtree-like classical percolation [Fig. 12(b)]. One observes thatfor the BFW model the size of the largest cluster grows dueto two smaller clusters merging together to become the newlargest (growth by overtaking), while in the classical case thelargest component mainly grows by adding small pieces [38].In the inset of Fig. 12(a) the symbols corresponding to thelargest cluster form the curve on top. When a jump occurs, thesize of the largest cluster changes by overtaking: for example,at the second to last jump shown in the inset, the second largestcluster merges with the third largest cluster and becomes thelargest cluster, while the previous largest cluster is then thesecond largest one. For the complete graph it has been shownthat all significant growth of the largest cluster size is due toovertaking [35].

We now analyze the behavior of the tree-like BFW model onhypercubic lattices. To estimate the percolation thresholds forthe infinite system, we consider the estimator pJ , i. e., the bondoccupation fraction at which the largest change in the order pa-rameter P∞ occurs (see Sec. II) and study its size dependence.For dimensions two to seven we obtain, respectively, pc =0.500 ± 0.001, 0.333 ± 0.002, 0.248 ± 0.002, 0.198 ± 0.003,0.165 ± 0.004, and 0.141 ± 0.009 (see Fig. 13). These values

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FIG. 12. (Color online) Fraction of sites in the largest cluster P∞as a function of the fraction of occupied bonds p for (a) the tree-likeBFW model and (b) tree-like classical percolation on hypercubiclattices of dimension two to seven (from right to left with increasingdimension). For the BFW model the transition is abrupt. In theclassical case, the transition is continuous and the thresholds arecompatible with results for the number of clusters at the criticalpoint [57–59]. The insets show the fraction of sites s/N in the threelargest clusters [1st (�, upper curve), 2nd (�, middle curve), and 3rd(◦, lower curve)] as a function of the fraction of occupied bonds p

for single typical realizations of (a) the tree-like BFW model and(b) tree-like classical percolation on the hypercubic lattice ofdimension seven of size 67. While for the BFW model the largestcluster grows by overtaking, i. e., the second largest cluster overtakesthe largest one which becomes the second largest one, for classicalpercolation the largest cluster grows directly by merging with smallerclusters. For the main plots, the system sizes Ld are 20482, 1613, 454,215, 136, and 97 for both models. Results have been averaged over atleast 2 × 103 samples.

are within error bars equal to 1/d. In Ref. [32] it was shown thatfor any percolation model on hypercubic lattices of dimensiond with finite number of clusters at the threshold pc � 1/d.If, in addition, the models are tree-like, pc = 1/d was shown.The values of the jump J are summarized in Fig. 14, where J

is plotted as a function of the system size N for the tree-likeBFW model in all considered dimensions. One observes thatthe jump value converges toward a constant value for largesystem sizes, which is a strong evidence of a discontinuoustransition [38].

In the following, we investigate the cluster-size distributionat the threshold for high dimensions. In Fig. 15 we see a plot ofthe number of clusters of size s per site ns as a function of thecluster size s for hypercubic lattices of dimension three, five,

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0.0000

0.0001

0.0002

0.0003

0.0004

10-9 10-7 10-5 10-3

p c−pc,J

N-1

2D3D4D5D6D7D

10-8

10-3

103 109N

FIG. 13. (Color online) Percolation threshold estimator pc − pc,J

as a function of the inverse system size N−1 for the tree-like BFWmodel on the hypercubic lattice of dimensions two to seven. Ex-trapolating to L → ∞, we find pc = 0.500 ± 0.001, 0.333 ± 0.002,0.248 ± 0.002, 0.198 ± 0.003, 0.165 ± 0.004, and 0.141 ± 0.009.Note that pc seems to be 1/d . In the inset we see the same datafor pc − pc,J , plotted as a function of N = Ld in double-logarithmicscale, where the solid line is a guide to the eye of the form 0.018N−0.7.Results have been averaged over at least 7.8 × 106 samples for thesmallest system size (642, 163, 84, 55, 46, and 47) and over at least100 samples for the largest one (81922, 5123, 914, 375, 206, and 167).

and seven. To reduce statistical fluctuations, we consideredthe accumulated number of clusters Ns , defined as Ns =∑2s−1

r=s nr . It can be observed that the cluster-size distributionis composed of two parts as the dimension increases, inagreement with the behavior on the random graph [34,35].The number of small clusters at the threshold increases withdimension. Thus, the cluster-size distribution for the tree-likeBFW model behaves in a different way than in classicalpercolation at the critical point, where asymptotically a powerlaw holds, ns ∼ s−τ or Ns ∼ s1−τ (see inset of Fig. 15).

Besides finding some evidence that the tree-like BFWmodel on the hypercubic lattice exhibits a discontinuous

0.405

0.410

0.415

0.420

0.425

0.430

103 105 107 109

J

N

2D3D4D5D6D7D

FIG. 14. (Color online) The largest change in the order parameter:J is plotted as a function of N = Ld for the tree-like BFW modelon hypercubic lattices of dimension two to seven. Asymptotically,the jump is independent on system size, indicating a stronglydiscontinuous transition. The solid lines are guides to the eye indi-cating the following constant values: 0.415 ± 0.005, 0.414 ± 0.002,0.412 ± 0.003, 0.410 ± 0.003, 0.409 ± 0.002, and 0.409 ± 0.002,for dimension two to seven.

10-10

10-8

10-6

10-4

10-2

100

102

100 102 104 106 108

Ns

s

5123

4013

345

285

137

107

10-9

100

100 109

Ns

s

slope:-187/91+1

FIG. 15. (Color online) Accumulated cluster-size distribution Ns

as a function of the cluster size s for the tree-like BFW model onhypercubic lattices of dimension three (�,�), five (�,�), and seven(�,�). Ns = ∑2s−1

r=s nr , where ns is the number of clusters of size s

per lattice site N = Ld , excluding the largest cluster of size smax. Oneobserves that the distribution falls off sharply for small cluster sizesand has a region on the right, qualitatively similar to the behavior onthe random graph [34,35]. For comparison, the inset shows Ns as afunction of s for tree-like classical percolation on the square latticeat pc = (7 − 3

√3)/4 [57–59]. Results are shown for system sizes of

81922 (◦) and 163842 (•). Asymptotically, the points follow a powerlaw, Ns ∼ s1−τ , where τ = 187/91. All results have been averagedover 100 samples.

transition, the results presented in this section also show thatplateaus in the evolution of the fraction of sites in the largestcluster P∞ on the simple-cubic lattice (see Fig. 8) are due toredundant bonds, which are not present in the tree-like case,in agreement with what has been reported for the randomgraph [34,35]. However, on the two-dimensional square lattice,the jumps occur at one unique fraction of occupied bonds,irrespective of whether the tree-like version is considered ornot. This result is in agreement with the results of Moreiraet al., showing that to obtain a discontinuous percolationtransition in two dimensions, the control of internal andredundant bonds is irrelevant [30].

V. FINAL REMARKS

We analyzed the BFW model [34–36] on the latticeand found numerical evidence for a strongly discontinuoustransition [38] in two and three dimensions. The surface fractaldimension of the BFW model in two dimensions was measuredto be df = 1.49 ± 0.02. Interestingly, this is different fromthe fractal dimension reported for another discontinuouspercolation model, 1.23 ± 0.03 [24,53], but similar to theone found in irreversible aggregation at high concentration,1.60 ± 0.08 [54]. The BFW model on the simple-cubic latticewas found to exhibit multiple distinct jumps of the largestcluster size in the thermodynamic limit. We further analyzedthe tree-like BFW model on hypercubic lattices of dimensiontwo to seven, finding that the jump size is asymptoticallyindependent on system size and determined the percolationthresholds. These results imply that the coexistence of multiplestable components in the BFW model is due to redundantbonds. This can be seen from the fact that the tree-like versionof the BFW model only exhibits one jump while multiple

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jumps at distinct fractions of occupied bonds appear onlyfor the simple-cubic lattice when redundant bonds are alsosampled. We also considered the evolution of the cluster-sizedistribution at the threshold with increasing dimension.

The results presented here are consistent with the behaviorof the BFW model on the random graph [34,35]. Themechanisms leading to discontinuous percolation in thiscase are compatible with the required ones according toRef. [30], namely, the homogenization of cluster sizes andthe control of merging bonds. Identifying percolation modelswith discontinuous transitions is interesting since the originalproposal [3] and related models have been shown, bothanalytically and numerically, to yield continuous transitionsin the thermodynamic limit [19]. Nevertheless, these initialmodels are characterized by a macroscopic jump for finitenetworks even with 1018 nodes [20,38]. Note that the Internetand the WWW have less than 1012 nodes.

It would be interesting to understand the finite-size scalingof discontinuous percolation models in a more rigorousanalytical manner since, in the presence of redundant bonds, itdiffers from the one of equilibrium first-order phase transitions[39]. In addition, interesting fractal dimensions emerged in theinvestigation of cluster perimeters in discontinuous percolation[32,44–52], which could also be studied for the BFW modelin dimensions higher than two.

ACKNOWLEDGMENTS

We acknowledge financial support from the ETH RiskCenter. We also acknowledge financial support from theBrazilian agencies CNPq, CAPES, and FUNCAP, the PronexGrant CNPq/FUNCAP, the Defense Threat Reduction AgencyAward No. HDTRA1-10-1-0088, and the Army ResearchLaboratory Award No. W911NF-09-2-0053.

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