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Unstable supercritical discontinuous percolation transitions

Wei Chen,1, 2, 3, ∗ Xueqi Cheng,1 Zhiming Zheng,4 Ning Ning Chung,5 Raissa M. D’Souza,3, † and Jan Nagler6, 7, ‡

1Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China2School of Mathematical Sciences, Peking University, Beijing, China

3University of California, Davis, California 95616, USA4Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education,

Beijing University of Aeronautics and Astronautics, 100191 Beijing, China5Department of Physics, National University of Singapore, 117542, Singapore

6Max Planck Institute for Dynamics and Self-Organization (MPI DS), Gottingen7Institute for Nonlinear Dynamics, Faculty of Physics, University of Gottingen, Gottingen, Germany

The location and nature of the percolation transition in random networks is a subject of intenseinterest. Recently, a series of graph evolution processes have been introduced that lead to discon-tinuous percolation transitions where the addition of a single edge causes the size of the largestcomponent to exhibit a significant macroscopic jump in the thermodynamic limit. These processescan have additional exotic behaviors, such as displaying a “Devil’s staircase” of discrete jumps inthe supercritical regime. Here we investigate whether the location of the largest jump coincides withthe percolation threshold for a range of processes, such as Erdos-Renyi percolation, percolation viaedge competition and via growth by overtaking. We find that the largest jump asymptotically oc-curs at the percolation transition for Erdos-Renyi and other processes exhibiting global continuity,including models exhibiting an ‘explosive’ transition. However, for percolation processes exhibit-ing genuine discontinuities, the behavior is substantially richer. In percolation models where theorder parameter exhibits a staircase, the largest discontinuity generically does not coincide withthe percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on themodel parameter. Distinct parameter regimes well in the supercritical regime feature unstable dis-continuous transitions—a novel and unexpected phenomenon in percolation. We thus demonstratethat seemingly and genuinely discontinuous percolation transitions can involve a rich behavior insupercriticality, a regime that has been largely ignored in percolation.

PACS numbers: 64.60.ah, 64.60.aq, 89.75.Hc, 02.50.Ey

I. INTRODUCTION

Percolation is a pervasive concept [1], which has ap-plications in a wide variety of natural, technological andsocial systems [2–5], ranging from conductivity of com-posite materials [6, 7] and polymerization [8] to epi-demic spreading [9–11] and information diffusion [12, 13].Viewed in a network setting, once the density of edgesexceeds a critical threshold, pc, the system undergoes asudden transition to global connectivity, where the size ofthe largest connected component transitions from micro-scopic to macroscopic in size. If, rather than edge density,we consider the impact of adding individual edges, weexpect to observe the largest jump in size of the largestcomponent at pc.

It is well known that the classic Erdos-Renyi (ER) [14]model of percolation undergoes a continuous second-

order phase transition during link-addition [15]. Here onestarts from a collection of N isolated nodes and edges areadded uniformly at random, with the critical edge den-sity, pc = 0.5. Instead of Erdos-Renyi, we can considercompetitive percolation processes [16]. In a competitive

∗Electronic address: chenwei2012@ict.ac.cn†Electronic address: raissa@cse.ucdavis.edu‡Electronic address: jan@nld.ds.mpg.de

process, rather than a single edge, a fixed number of edges(or nodes) are chosen uniformly at random, but only theedge that best fits some specified criteria is added to thegraph. Competition between edges is typically referredto as an Achlioptas Process [17].

Achlioptas Processes (AP) can exhibit a very sharp ex-

plosive transition which appears discontinuous on any fi-nite system [17]. In past years such sharp transitions havebeen demonstrated for scale-free networks [18–20], squarelattices [21, 22], Bethe lattice [23], directed networks [24]and more realistic systems [25–27]. Although strong nu-merical evidence suggests that many explosive AP arediscontinuous [17], more recently it has been shown thatthe seeming discontinuity at the percolation transitionpoint disappears in the thermodynamic limit [16, 28–32].However, this neither means that all AP are necessarilyglobally continuous nor that there are no genuine discon-tinuities during the first continuous emergence of a giantcomponent. In fact, a giant connected component canemerge in a series of infinitely many genuinely discontin-uous jumps and the notion that explosive percolation isalways continuous [30] is thus misleading [33].

Several random network percolation models have nowbeen identified and studied that show a single gen-uine discontinuous transition [29, 34–36, 38–45], or evenmultiple discontinuities [33, 37, 46]. The mechanismsfor discontinuous transitions such as dominant overtak-ing [16, 47], cooperative phenomena [42] and the suppres-

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t c)/N

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FIG. 1: Continuous percolation models. (a) A typical evolution of C1 as a function of the link density p = t/N for ER,and for the PRmax, PRmin models with the number of candidate edges c = 2 and c = 4, respectively. The system size N = 106.(b) (t∆ − tc)/N versus number of nodes N for these models, all of which follow a power-law distribution. Each data point in(b) is averaged over 1000 realizations.

sion principle [48], have received considerable attention,as have criteria to discriminate between continuous anddiscontinuous explosive percolation transitions. A signa-ture for a continuous percolation transition is a criticalpower-law component-size distribution [20, 28] and anasymptotically vanishing order parameter at the phasetransition point [36].

A method to discriminate between weakly and gen-uinely discontinuous transitions proposed in [16] is to usethe asymptotic size of the largest jump in the order pa-rameter from the addition of a single edge. Importantly,if the largest jump of the order parameter does not van-ish as the system size N → ∞, the transition is neces-sarily discontinuous. However, whether the largest jumpin the order parameter asymptotically coincides with thepercolation transition point, and thus announces it, hasremained largely unaddressed.

In this paper, we study whether the position of thelargest jump in the order parameter asymptotically con-verges to the percolation transition point. To exem-plify this we study the Erdos-Renyi model, AchlioptasProcesses and the generalized Bohman-Frieze-Wormaldmodel (BFW) [49]. We find that the position of thelargest jump in the order parameter asymptotically con-verges to the percolation transition point for ER and APwith global continuity, but not necessarily for AP withdiscontinuities. For the BFW model, it depends on thevalue of the model parameter α. In BFW multiple gi-ant components emerge at the percolation point, withthe value of α determining the number of giants [35].Here we show that there are further sub-regimes of αvalues, with “stable” regions where the macroscopic com-ponents never merge and “unstable” regions where giantscan have further merging in the supercritical regime. In

stable α regions the largest jump coincides with the per-colation threshold, but in unstable regions the largestjump is in the supercritical regime.

II. PERCOLATION MODELS WITH GLOBALCONTINUITY

We study whether the position of the largest jump inthe order parameter asymptotically converges to the per-colation transition point for percolation models exhibit-ing global continuity. The best understood percolationmodel that shows a continuous phase transition is theErdos-Renyi model (ER) [14]. Let N be the number ofnodes, t be the number of links (i.e., edges) in systemand let Ci denote the fraction of nodes in the i-th largestcomponent. A typical evolution of C1, as a function ofthe link density p = t/N (number of links per node) isshown in Fig. 1 (a). To study whether the position ofthe largest jump in C1 converges to the percolation tran-sition point, we measure (t∆ − tc)/N as a function of N ,where tc denotes the minimal number of steps for C1 toexceed N1/2, and t∆ is the time (number of steps) whenthe largest jump in C1 has occurred. Fig. 1 shows that(t∆ − tc)/N ∼ N−0.227, which suggests that (t∆ − tc)/Nasymptotically converges to zero. In addition both t∆/Nand tc/N converge to the percolation transition point inthe thermodynamic limit.Next we study two extremal AP models, specifically

the PRmax model and the ‘explosive’ PRmin model. Inthe PRmax model, two candidate links are selected ran-domly at each step and the link that maximizes the prod-uct of the component sizes that the ends of the link residein is added while the other link is discarded. In the PRmin

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FIG. 2: Unstable supercritical discontinuous transitions. Typical evolutions of C1 and C2 versus the link density forthe DS model (a), NG model (b) and mER model (c) with system size N = 106. The link density p1 at which the largest jumpin C1 occurs, the link density p2 when the second largest jump in C1 occurs and p1 − p2 versus system size N for the DS model(d), NG model (e) and mER model (f). Each data point in (d), (e) and (f) is averaged over 100 realizations.

model, c candidate links are selected randomly at eachstep and the link that minimizes the product of the com-ponent sizes that the ends of the link reside in is addedwhile all other links are discarded. As an example, inFig. 1(a) we show the evolution of C1 as a function ofthe link density p = t/N for PRmax and PRmin. Simi-larily to the numerical results for the ER model, we findthat (t∆ − tc)/N follows a power-law distribution for allstudied models. In particular (t∆ − tc)/N ∼ N−0.223 forPRmax, ∼ N−0.362 for PRmin with c = 2, and ∼ N−0.449

for PRmin with c = 4.This suggests that in continuous percolation, even if

extremely abrupt, the largest gap asymptotically an-nounces the percolation transition. This convergence,however, is not always guaranteed as we will show next.

III. MODELS WITH DISCONTINUITIES

In contrast to AP models discussed in the previous sec-tion, several random neighbor models (of the AP class)have been studied that exhibit ‘staircase’ discontinuitiesin the supercritical regime [33, 50]. Here we focus onthree types of AP models, the Devil’s staircase model(DS) [33], the Nagler-Gutch model (NG) [50], and themodified ER model (mER model) [50]. The DS model

is based on picking three nodes at random and forbid-ding the largest picked component to merge with com-ponents whose sizes are not similar, which results in aDevil’s staircase with an infinite hierarchy of discontin-uous jumps in C1 [33], see Fig. 2 (a). This model hasbeen used as a counterexample for the conclusion madein Ref. [30] that explosive percolation should be alwayscontinuous [33].

Like the DS model, the NG model and mER model areboth based on 3-node rules in which the addition of linksconnecting two components whose sizes are similar is fa-vored. Let p1 denote the link density immediately afterthe largest jump in C1 from the addition of a single edge,and p2 denote the link density immediately after the sec-ond largest jump in C1 from the addition of a single edge.Fig. 2 (d) shows that p1, p2 and p1 − p2 asymptoticallyconverge to some positive constant as N → ∞, becauseof the occurrence of multiple discontinuous jumps in thesupercritical regime. Figures 2 (b) and (c) show realiza-tions of C1 and C2 for the NG and the mER model, inboth cases featuring multiple discontinuous transitions ofC1 [50]. Figures 2 (e) and (f) show that for both the NGmodel and the mER model, p1, p2 and p1−p2 asymptot-ically converge to some non-zero constant as the systemsize N increases. This numericial result can be under-stood in the following way. From numerical observations,

4

the size of the third largest component immediately be-fore p2 in the supercritical regime is at most O(logN) forthe DS model, the NG model and mER model. In addi-tion, for all these models it has been analytically demon-strated that immediately before p2 both the size of thelargest and second largest component are of order O(N),where for the DS model it can be shown that C2 = 1

2C1,

while for the NG model and the mER model C2 = C1

[33, 50]. As a result, for these models once the largestand second largest component merge together inducing adiscontinuous jump, the size of the second largest com-ponent drops to O(logN). Thus O(N) links are requiredbefore at p1 the second largest component grows againto size 1

2C1 (for the DS model), or to size C1 for the NG

model and the mER model, respectively. Thus, p1 − p2is necessarily extensive and asymptotically converges tosome positive non-zero constant.This demonstrates the occurrence of multiple discon-

tinuous transitions, including the transition with thelargest discontinuity, in the supercritical regime and notat the percolation critical point as in traditional percola-tion.In general, we use the term stable coexistence when all

giant components emerging at the percolation transitionpoint persist and remain separate throughout the super-critical regime. We use the term unstable coexistencewhen at least two giant components emerging at the per-colation transition point merge together at some pointin the supercritical regime. We find that all the modelsstudied in this section display unstable supercritical dis-continuous transitions, which is a novel and unexpectedfeature in percolation. The model we study next showseven a quantitatively richer behavior, with some regionsof stable coexistence and other regions of unstable coex-istence.

IV. BOHMAN-FRIEZE-WORMALD MODEL

The Bohman-Frieze-Wormald Model was first intro-duced to demonstrate that the emergence of a giant com-ponent can be largely suppressed [49]. (The details ofBFWmodel can be found in the appendix.) In particular,it has been established that under the BFW evolution,if m = 0.96689N links out of 2m sequentially sampledrandom links have been added to a graph, a giant com-ponent does not exist [49]. More recently, the nature ofthe BFW transition was investigated. It was establishedthat multiple giant components appear in a discontinu-ous percolation transition for the BFW model [35]. Inthe asymptotic time limit of the BFW model, one-halfof all edges that are sampled must be added. Generaliz-ing the BFW model by allowing the asymptotic fractionof edges to be a parameter α, then the number of giantcomponents can be tuned by adjusting the value of α [35].For this model we investigate whether discontinuous

jumps of the order parameter occur at the percolationtransition point or in the supercritical regime. First we

study the evolution of the size of the four largest com-ponents, Ci with i = 1, 2, 3, 4 for the BFW model withα ∈ (0, 1], as a function of the link density p.

In Fig. 3(a) we show the discontinuous emergence of aunique giant component in a sharp transition to globalconnectivity, for α = 0.6. For other values of α, multiplegiant components emerge simultaneously in a sharp tran-sition. See, for instance, Fig. 3(c) for α = 0.5 with two gi-ant components, or Fig. 3(e) for α = 0.3 with three giantcomponents. Thus, for certain values of α, there existsa unique transition to global connectivity with multiplegiant components.

Yet, Figs. 3(b), (d), (f) show there is another type ofbehavior possible, with an additional transition in the su-percritical regime where giant components merge. Thissuggests the existence of an instability of giant compo-nents. Next, we perform numerical simulations to charac-terize the occurrence of discontinuities and multiplicitiesduring the discontinuous transitions.

As in the previous section, we denote the link densityp1 as the position immediately after the largest jump inC1 from the addition of a single edge, and p2 the po-sition immediately after the second largest jump in C1

from the addition of a single edge. In addition, let p3denote the minimal position at which the largest com-ponent contains at least N1/2 nodes. From numericalresults in Fig. 3, we find that p1 ≥ p2 ≥ p3.

Let us focus initially on the region with stable coex-isting giant components. For α = 0.6, 0.5, 0.3, Fig. 4 (a),(c), (e) show that the size of the largest component at p1,denoted as C1(p1), is almost independent of the systemsize N and converges to some positive constant asymp-totically. On the other hand, the gap between p1 and p3,i.e. ∆p = p1 − p3, scales as a negative power of N andthus decreases to zero as N → ∞, see Fig. 4. This sug-gests that once the number of nodes in the largest com-ponent increases from O(N1/2) to O(N), the augmentedlink density converges to zero as N → ∞, indicatingthe percolation process undergoes a unique discontinu-ous transition in the thermodynamic limit.

We further find that at p1, for α = 0.5, C2 convergesto some positive constant as well, see Fig. 4 (c), and forα = 0.3, C1, C2, and C3 all converge to some positivenon-zero constant, respectively, see Fig. 4 (e). This indi-cates that the multiple giant components appear simulta-neously in a unique discontinuous transition, consistentwith the theory and observations put forth in [35, 47].

Let us now focus on the regime with unstable coex-istence of multiple giant components. For α in the un-stable regime, the size of the largest component at p2,is almost independent of the system size N and con-verges to some positive constant asymptotically. Forα = 0.53, 0.35, 0.26, the size of the two, three and fourlargest components at p2 converge to some positive con-stant asymptotically. Yet, we find that p2−p3 decays as apower law in N , see Fig. 4 (b), (d), (f). This suggests thepercolation transition is discontinuous at p2 with multi-ple giant components emerging simultaneously.

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FIG. 3: Critical and unstable supercritical discontinuous transitions. (a) For α = 0.600, one giant componentemerges in a phase transition. (b) For α = 0.530, C1, C2 versus the link density, showing that two giant components emergesimultaneously in the first phase transition. They are however unstable as they merge at a second transition. (c) For α =0.500, C1, C2, C3 versus the link density, showing that two giant components emerge simultaneously. (d) For α = 0.350,C1, C2, C3 versus the link density, showing that three giant components emerge simultaneously in the first phase transition.This configuration is unstable as in a second transition the second largest and the third largest components merge. (e) Forα = 0.300, C1, C2, C3, C4 versus link density, showing the simultaneous emergence of three giant components. (f) For α = 0.260,C1, C2, C3, C4 versus density of links, showing that four giant components emerge simultaneously in the first phase transition.Two smallest macroscopic components C3 and C4 merge together and overtake C1, the other macroscopic components arestable in the remaining process, C3 + C4 → C1, C1 → C2, C2 → C3. System size is N = 106 for all simulations.

In addition, we observe that the size of the largest jumpof the largest component at p1, denoted by max∆C1, isindependent of system size N and converges to some pos-itive constant asymptotically (see Fig. 4 (b), (d), (f)).This suggests the occurrence of a second discontinuoustransition at p1 in the supercritical regime. We findthat the second transition results from the merging ofthe two smallest giant components that emerge at the

first (i.e., percolation) transition. This mechanism canbe seen clearly from the case of α = 0.26 in Fig 3 (f),where four giant components emerge at the first tran-sition while at the second transition, C3 and C4 mergetogether and overtake C1 in size. Thus C1, C2, C3 all geta sudden jump in size but C4 breaks down. This over-taking mechanism dominates the growth of the largestcomponent in the BFW model, which has been proven

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FIG. 4: Scaling of critical and unstable supercritical discontinuous transitions. (a) For α = 0.600, C1 at p1, p1 − p3versus system size. (b) For α = 0.530, C1 at p2, the largest jump in C1, p2 − p3, p1 − p2 versus system size. (c) For α = 0.500,C1, C2 at p1, p1 − p3 versus system size. (d) For α = 0.350, C1, C2 at p2, the largest jump in C1, C2, p2 − p3, p1 − p2 versussystem size. (e) For α = 0.300, C1, C2, C3 at p1, p1 − p3 versus system size. (f) For α = 0.260, C1, C2, C3 at p2, the largestjump in C1, C2, C3, p2 − p3, p1 − p2, versus system size. Each data point is averaged over 1000 realizations.

to be a key mechanism leading to discontinuous percola-tion transitions [16, 33, 47].

To test if the transition points p1 and p2 in the unsta-ble regime are still distinct in the thermodynamic limit,we next perform a scaling analysis. For the values ofα = 0.53, 0.35, 0.26, which are in the unstable region, wefind that p1 − p2 converges to a non-zero constant, seeFig. 4. This suggests, indeed, the distinctness of the twotransition points.

Taken together, we have identified and studied two pa-rameter regimes in the BFM model, (i) the stable regimeof a unique discontinuous transition where one or more

giant components emerge and coexist throughout the su-percritical regime. (ii) the unstable regime of multiplediscontinuous transitions where multiple giant compo-nents emerge but the two smallest ones merge at a well-defined transition point in the supercritical regime.

However, the characterization remains incomplete asso far we have only studied three instances for each pa-rameter regime. Our next aim is to establish a phasediagram by continuously tuning the parameter α.

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FIG. 5: Phase diagram for the BFW model. The number of giant components that appear in the first discontinuousphase transition (percolation transition point) in dependence on α. In the stable region, the giant components remain distinctin the supercritical regime while in the unstable region the two smallest giant components merge in the supercritical regime ata well-defined transition point.

Phase Diagram of the BFW Model

We first investigate the behavior in the regime α > α1

with α1 = 0.511±0.003, where only one giant componentasymptotically remains in the system. For values α < α1,two giant components asymptotically remain [35].

Since in a stable regime of α all giant components thathave emerged remain separate throughout the supercrit-ical regime, a stable regime is characterized by a uniquetransition of the largest component. In contrast, an un-stable regime is characterized by one (or more) discon-tinuous transitions of C1 in the supercritical regime byaggregation of two (or more) giant components emergingat percolation transition point.

We find that the model undergoes two distinct discon-tinuous phase transitions for α ∈ (α1, α

′1), referred to as

the unstable regime, but undergoes a unique discontin-uous phase transition for α ∈ (α′

1, 1), referred to as thestable regime, where α′

1 = 0.551± 0.001. Previous workhas established an infinite number bifurcation points αi,at which the number of stable giant components thatasymptotically remain in the system changes from i toi + 1, i ≥ 1 (see table. I) [35]. Here we numericallyexpand the analysis including transitions of the secondlargest component, the third largest component, and theforth largest component as well. These transitions leadto multiple discontinuous transitions of the largest com-ponent and a hierarchy of stable and unstable regimes. Inparticular, for α ∈ (α′

i, αi−1), i ≥ 2, we identify a stable

TABLE I: Summary of bifurcation points αi and α′i.

i αi α′i

1 0.511 ± 0.003 0.551 ± 0.001

2 0.343 ± 0.001 0.352 ± 0.001

3 0.259 ± 0.001 0.261 ± 0.001

4 0.206 ± 0.001 0.208 ± 0.001

regime, where in a unique discontinuous transition the ilargest macroscopic components, C1, C2, ..., Ci, simulta-neously emerge, and for α ∈ (αi, α

′i) an unstable regime

with two distinct discontinuous transitions, for i ≥ 1.We find the numerical values α′

2 = 0.352 ± 0.001, α′3 =

0.261± 0.001, α′4 = 0.208± 0.001..., see table I.

In Fig. 5 we demonstrate the phase diagram of theBFW model. We show the number of giant componentsthat emerge at the first transition (percolation transitionpoint) versus α, with alternating stable and unstable re-gions. Since the number of giant components that asymp-totically remain in the system increases as α decreases,there exist infinitely many bifurcation points α′

i, i ≥ 1,which separate stable from unstable regimes.

V. SUMMARY AND DISCUSSIONS

For various models with continuous, discontinuous andmultiple-discontinuous percolation transitions, we have

8

investigated whether the location of the largest jump inthe order parameter asymptotically converges to the per-colation transition point marking the onset of global con-nectivity.

For globally continuous transitions, including certainAchlioptas processes, the location of the largest jumpin the order parameter asymptotically converges to thepercolation transition point. In contrast, Achlioptas pro-cesses with discontinuities exhibit a ‘staircase’ of disconti-nuities in the supercritical region and the location of thelargest jump is at an edge density well above the per-colation point. Finally, the BFW model exhibits a richsupercritical behavior that is dependent on the model pa-rameter α, as exemplified by the phase diagram, Fig. 5,together with analytics suggesting an infinite hierarchy ofregimes of alternating stability type. Whether the per-colation transition is asymptotically announced by thelargest gap in the size of the largest component dependson the parameter. In the stable regime macroscopic com-ponents robustly coexist, displaying the largest jump ofthe order parameter at the percolation transition point.In the unstable region the coexistence of all macroscopiccomponents that have emerged occurs in a finite sizedwindow only, leading to multiple discontinuous transi-tions. Macroscopic components that emerge at the per-colation transition are thus not necessarily stable in thethermodynamic limit.

For AP models with discontinuities and for the BFWmodel, multiple discontinuous transitions are a conse-quence of the occurrence of extended periods in timewell in the supercritical regime where macroscopic com-ponents cannot merge. Mechanisms implying such peri-ods are yet to be discovered.

Multiple transitions have been studied in a wide varietyof fields, such as geophysics [51], liquid crystals [52], clas-sical thermodynamics and solid state physics [53]. How-ever, in random network percolation multiple transitionsare poorly understood. It would thus be interesting toidentify the sufficient conditions for these [33, 55]. Thisnumerical work represents a step towards this direction.

In short, we have investigated unstable discontinuoustransitions in percolation. Seemingly and genuinely dis-continuous percolation transition can involve a rich be-havior in supercriticality, a regime that deserves atten-tion for percolating systems with substantial delays [55].

Acknowledgments

This work was supported by the Defense Threat Re-duction Agency, Basic Research Award No. HDTRA1-10-1-0088, the Army Research Laboratory under Co-operative Agreement Number W911NF-09-2-0053, the973 National Basic Research Program of China (No.2013CB329602, 2012CB316303), the National NaturalScience Foundation of China (No. 11305219).

Appendix: Algorithm of percolation models

We state the algorithm of the percolation models stud-ied in this paper in detail, which are the Devil’s stair-case model, the Nagler-Gutch model, the modified Erdos-Renyi model and the Bohman-Frieze-Wormald model.

1. Devil’s staircase (DS) Model

Start with an empty graph with N isolated nodes. Ateach step, three nodes v1, v2, v3 are randomly selectedfrom N nodes, and let s1, s2, s3 denote the sizes ofcomponents (not necessarily distinct) in which theyreside. Consider ∆i,j = |si − sj | with 1 ≤ i < j ≤ 3 andconnect vi, vj for which ∆i,j is minimal.

2. Nagler-Gutch (NG) Model

Start with an empty graph with N isolated nodes. Ateach step, three nodes v1, v2, v3 are randomly selectedfrom N nodes, and let s1, s2, s3 denote the sizes of com-ponents (not necessarily distinct) in which they reside.Let si denote the size of the component containingvi. If all three component sizes si are equal, add theedge connecting v1v2. If exactly two component sizes siare equal, connect the corresponding nodes by a link.Otherwise (if all si are different), link the nodes in thetwo smallest components. This model is a modificationof the “explosive” triangle rule introduced in [54], whichin contrast exhibits a steep but continuous transition.

3. Modified Erdos-Renyi (mER) Model

Let L1 and L2 denote the sizes of the two largestcomponents of the evolving graph. The mER modelproceeds as follows. If the two largest components in thecurrent graph have the same size (L1 = L2), add the edgeconnecting v1v2. When L1 > L2, if at least two si areequal to L1, connect two corresponding nodes, otherwiseconnect two nodes in components of size smaller than L1.

4. Bohman-Frieze-Wormald (BFW) Model

Start with an empty graph with N isolated nodes andproceed in phases starting with phase k = 2. Edges aresampled one-at-a-time, uniformly at random from thecomplete graph. If an edge would lead to formation of acomponent of size less than or equal to k it is accepted.Otherwise the edge is rejected provided that the fractionof accepted edges is greater than or equal to a function

9

g(k) that decreases with k. If the accepted fraction isnot sufficiently large, the phase is augmented to k + 1repeatedly until either the edge can be accommodated org(k) decreases sufficiently that the edge can be rejected.Stated formally, let k denote the stage, N the number

of nodes, u denote the total number of links sampled andA the set of accepted links (initially A = ∅), and t = |A|the number of accepted links. At each step u, a link eu issampled uniformly at random from the complete graphgenerated by the N nodes, and the following algorithmiterated:

Set l = maximum size component in A ∪ {eu}

if (l ≤ k) {

A← A ∪ {eu}

u← u+ 1 }

else if (t/u < g(k)) { k ← k + 1 }

else { u← u+ 1 }

where g(k) = α+√

1/2k with α ∈ [0, 1]. Thus α denotesthe asymptotic fraction of accepted links over totallysampled links. In the original BFW model, α = 1/2 [49].We generalize the BFW model by tuning α.

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