Construction and Analysis of Random Networks with Explosive Percolation
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Construction and Analysis of Random Networks with Explosive Percolation
Eric J. Friedman and Adam S. Landsberg School of ORIE and Center for Applied Mathematics, Cornell University, Ithaca, NY 14850. Joint Science Department, Claremont McKenna, Pitzer, and Scripps Colleges, Claremont, CA 91711 (Dated: October 29, 2018)The existence of explosive phase transitions in random (Erd˝os R´enyi-type) networks has beenrecently documented by Achlioptas et al. [Science , 1453 (2009)] via simulations. In this Letterwe describe the underlying mechanism behind these first-order phase transitions and develop toolsthat allow us to identify (and predict) when a random network will exhibit an explosive transition.Several interesting new models displaying explosive transitions are also presented.
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The structure and dynamics of networked models andtheir application to social networks is an important andactive area of research encompassing many fields rang-ing from physics [1, 2, 3] to sociology [4] to combina-tions thereof [5, 6, 7]. Ideas from statistical mechanicshave contributed greatly to our understanding of suchnetworks and their practical uses [8, 9, 10]. Of partic-ular importance is the statistical mechanical notion ofthe order of a phase (‘percolation’) transition. Phasetransitions in random network models are almost alwayssecond order [11, 12] or higher [12, 13]. Thus, it wassurprising to many when Achlioptas et al. [14] reportedrecently that some models of interest in social networkscan display first-order (discontinuous) transitions.In that work, they described several random graphmodels of the Erd˝os R´enyi (ER) variety that exhibit first-order or what they call “explosive” phase transitions.They provide convincing numerical evidence and a usefulcharacterization of such transitions, but no details on themechanisms underlying them. They describe several sys-tems which display such transitions and a general classof systems which don’t.In this paper we describe the underlying mechanismsbehind explosive transitions in ER-type models. Weshow that, somewhat surprisingly, the key to explosivetransitions is not the details of the edge-addition rules atwork during the actual “explosion,” but rather lies in theperiod preceding the explosion when a type of “powderkeg” develops. In effect, the importance of the rules isto create an explosive situation, which can be detonatedwith almost any rule. In addition, our analysis providesan understanding of which random network models willhave such transitions. This allows us to construct largeclasses of interesting models that display this behavior.(It also allows us to rule out many other models whichwill not display explosive transitions.)The prototypical network percolation example is thatof pure (non-preferential) ER random graphs [11]. Thesebegin with a set of n nodes, where n is large. Edgesare then added to the graph, uniformly at random. As iswell known, this system exhibits a phase transition as thenumber of edges τ increases. For τ < . n all clusters aresmall ( ∼ log ( n )) while for τ > . n a large cluster ( ∼ n ) τ /n s ( τ ) / n LCERMCMC3SC
FIG. 1: Transition diagrams for largest cluster (LC), Erdos-Renyi (ER), min-cluster (MC), min-cluster 3 (MC3) andsmallest cluster (SC) rules. LC and ER display second-ordertransitions while the others are first order. appears. In the large- n limit this transition is a second-order phase transition, i.e., letting s ( τ ) be the size of thelargest cluster after τ edges have been added, the graphof s ( τ ) /n against τ /n is continuous, as seen in Figure 1.Achlioptas et al. [14] considered a variety of ER-likerandom networks using modified edge-addition proce-dures, wherein, at every step, two candidate edges arechosen at random, but only one of the two is actuallyadded to the graph. Under the min-cluster (MC) rule,for example, one selects the edge which minimizes thesum of the weights of the nodes in the edge, therebycreating the smaller cluster. (Here, the “weight” of anode is defined as the size of the connected component(cluster) which contains that node.) As mentioned in[14], this rule leads to explosive phase transitions, as ev-idenced by the discontinuity in a plot of s ( τ ) /n against τ /n (Figure 1). To demonstrate that such transitionsare truly first order (discontinuous), they use the follow-ing approach: Letting t ( a ) be the ‘time’ (i.e., number ofedges added) when a cluster of size ≥ a first arises, de-fine ∆ = t ( n/ − t ( n α ), for some 0 < α <
1. Then, toshow that a transition is explosive they demonstrate, viasimulations, that ∆ ∼ n β for some β <
11 12 13 14 157891011121314 ln(n) l n ( ∆ ) ERMCMC3 ln( ∆ ) ≈ ∆ ) ≈ ∆ ) ≈ FIG. 2: A log-log plot of ∆ vs. n for ER, MC and MC3 rules,for α = 1 /
2. (Markers are for individual realizations.) width of the transition region in the rescaled network is∆ /n ∼ n β − , which vanishes in the large n limit. For theMC model, when α = 1 / β ≈ .
6, as shownin Figure 2. (Note that the condition β < F ( τ, a ) be the number of nodes in clustersof size ≥ a after the addition of the τ ’th edge. We findthat, under the MC rule,1 n F ( t ( n α ) , n − β )approaches a non-zero constant in the large- n limit, asshown in Figure 3 for α = 1 / β = 0 .
6. In otherwords, at time t ( n α ), which is the beginning of the phasetransition, a fixed (non-zero) fraction of nodes are con-tained in small clusters with sizes ranging from n − β to n α . The set of clusters in this size range constituteswhat we term the “powder keg.” Although each individ-ual cluster in the powder keg contains only a vanishinglysmall fraction of nodes, they are ignitable and collectively they enable an explosion to occur. Note that an analo-gous plot for the ER rule (Figure 3) shows that in that F ( t ( n . ) , n . ) / n MC ER F(t(n ),n )/n ≈ −3*10 −9 n + 0.02F(t(n ),n )/n ≈ −9 n + 0.34 FIG. 3: Plot of n F ( t ( n . ) , n . ) as a function of n for theErdos-Renyi (ER) and min-cluster (MC) rules. All coeffi-cients of regression are statistically significant ( p < − ). case the powder keg is empty in the large n limit.To see why the existence of a powder keg guaranteesan eventual explosion we first make a simple observation:If there is no cluster of size ≥ n/ / /
4. Consequently, we see that internal edges in thegraph will not impact the order of the phase transitionitself, and hence can be safely ignored. In light of this, wedefine ˆ τ to be a measure of the number of “non-internal”edges added to the graph. In general if there are C clus-ters in total in a graph at non-internal time ˆ τ , then at(non-internal) time ˆ τ + C − ≥ n − β which exist at time t ( n α ), i.e., clusters inthe powder keg. Note that there can be at most n β suchclusters, and that a finite (non-zero) fraction of nodesin the network belong to the powder keg, as shown inFigure 3. Thus, the probability of choosing an edge thatconnects two different clusters, both in the powder keg , isstrictly greater than 0. It therefore follows that the timefrom the creation of the powder keg, t ( n α ), to the timewhen a cluster of size n/ t ( n/ ∼ n β , since a positive fixed fraction of the edges addedjoined clusters from the powder keg. Thus∆ = t ( n/ − t ( n α ) ∼ n β . So once a powder keg is created, any reasonable edge-addition rule will eventually detonate it – i.e., the exis-tence of a powder keg guarantees an eventual explosivetransition in the network .To better understand this difference between non-explosive models (e.g., pure ER) and explosive mod-els (e.g., MC, PR), it is helpful to make note of twovery extreme, highly simplified models for which analyt-ical calculations are possible. Surprisingly, despite theirsimplicity, these capture many of the essential distinc-tions between second- and first-order transitions in net-works. In the so-called largest cluster (LC) model, ateach step one identifies the two largest clusters in thenetwork and adds an edge between them. In this casethe dynamics is characterized by a single cluster grow-ing over time, and a straightforward computation showsthat ∆ ≈ ( n/ − − ( n / − n the transition will be non-explosive. (See Figure 1.) Inthe smallest cluster (SC) model, at each step one identi-fies the two smallest clusters in the network and adds anedge between them. In this case a calculation shows that t ( a ) ≈ n (1 − /a ) , since the first n/ n/ n/ n/ ≈ ( n − − ( n − n / ) . Hence ∆ ∼ n / , which is explosive. In fact the SC rule isessentially the “most explosive” rule and more explosivethan the MC or PR rules. Its (rescaled) graph is simplya step function at 1. (See Figure 1.) Note that in thismodel the powder keg is extreme, as all the nodes are al-ways in the interval [ n α / , n α ] at t ( n α ), if n α is a powerof 2. This implies that α + β = 1, whereas α + β > ≥ a . Recall that under the MC rule, oneconsiders two potential edges (randomly chosen) and se-lects the one leading to the smaller resultant cluster size.So in order to form a cluster of size ≥ a , both these po-tential edges must contain at least one node of size ≥ a/ a ). Hence the probabilitythat the addition of a new edge to a graph at time τ willproduce a cluster of size ≥ a is at most 4( F ( τ, a/ /n ) to lowest order, since F ( τ, a/ /n is the probability ofrandomly choosing a node of size ≥ a/
2. (Formally, thefunction F is a random variable defined on sample paths.In our analysis, the bounds on F are all high probabil-ity bounds, i.e., the probability of them not holding willvanish as n → ∞ .)We next establish a lower bound for the quantity F ( t ( a ) , a/ ≥ a/ t ( a ) when the firstcluster of size ≥ a appears. First, observe that the ex-pected number of clusters of size ≥ a at this time (up toconstants and higher-order terms) is at most t ( a ) X i =0 F ( i, a/ /n ) which is bounded by 4 t ( a )( F ( t ( a ) , a/ /n ) because F ( u, a/
2) is non-decreasing in u . Next, we can write thisbound as 4 n ( F ( t ( a ) , a/ /n ) since the previous argu-ment concerning non-internal time shows that only values of time which are less than n need be considered. Nowsuppose (incorrectly) that F ( t ( a ) , a/
2) were less than n / − ǫ for some ǫ >
0. Then the expected number ofclusters of size ≥ a at time t ( a ) would be vanishinglysmall in the large- n limit ( ∼ n − ǫ ). This contradicts themeaning of t ( a ) as the time marking the first appearanceof a cluster of size ≥ a in the network. Hence, at thetime of the creation of the first cluster of size a , it mustbe the case that F ( t ( a ) , a/
2) is of order n / or greater.This bound is essential for understanding the creationof the powder keg under the MC and similar rules. Inessence, as a lower bound it underpins the build-up of asufficient number of clusters in the size range that consti-tutes the powder keg. We note that the analogous com-putation for the ER rule yields no corresponding lowerbound on F ( t ( a ) , a/ ǫ ’s fromthe analysis, since they can be chosen to be arbitrar-ily small.) Consider the situation when a cluster of size n α first forms, for α ≤ /
2. We just showed that, atthis time, the number of nodes F ( t ( n α ) , n α /
2) is of or-der n / or greater, which implies that there are at leastorder n / − α clusters of size ∼ n α . Now, the expectednumber of such clusters is bounded by n / − α ≈ t ( n α ) X i =0 F ( i, a/ /n ) ≤ n ( F ( t ( n α ) , a/ /n ) which implies that F ( t ( n α ) , n α /
4) is order n / − α/ . Iter-ating this argument shows that F ( t ( n α ) , n α / k ) is order n − α +(2 α − / k . This result hints at the creation of thepowder keg, but is not sufficient to prove its existence.Improving this argument requires a more detailed anal-ysis of the dynamics of the system. The following argu-ment is heuristic and instructive. (It remains an openproblem to formalize the argument.) Observe that noclusters of size n α can appear until at least one clusterof size n α / n α can be refined to t ( n α ) X i =0 ( F ( i, n α / /n ) = t ( n α ) X i = t ( n α / ( F ( i, n α / /n ) which is bounded by[ t ( n α ) − t ( n α / F ( t ( n α ) , n α / /n ) . Unfortunately, theanalytic computation of [ t ( n α ) − t ( n α / n β , where β = 1 − α , so the expected num-ber of clusters is n − α ( F ( t ( n α ) , n α / /n ) which yields F ( t ( n α ) , n α / ∼ n α . Iterating this argument givesthe bound F ( t ( n α ) , n α / k ) ∼ n − − α k . For k suitablylarge this indicates that the powder keg can containenough mass.Our analysis provides the foundations on which to un-derstand other ER-like models with explosive transitions.The key insight is the equation for the probability ofcreating a large cluster from smaller ones. In particu-lar, we conjecture that any “reasonable” network modelfor which the probability of creating a cluster of size a at time t has probability proportional to ( F ( t, a/ /n ) p with p > k randomly chosen candidateedges and selecting the one with the lightest node sum.When k = 1 this is the ER model, which is non-explosivesince p = 1. For k = 2 this is the MC model whichis explosive ( p > k these models are in-creasingly explosive and are denoted min-cluster- k rules.For example, in Figure 1 the min-cluster-3 rule (MC3)appears “more explosive” than the MC rule (though Fig-ure 2’s plot of ∆ ∼ n β is not numerically accurate enoughto firmly establish that MC3 truly has a smaller β valuethan that of MC). We can also extend this class of modelsto choosing the m’th lightest out of k for m ≤ k (hence-forth called the ( m, k ) models). Our analysis suggeststhat for m = 1 these models are not explosive, but for m > p > k .One of the limitations of the explosive network mod-els considered by Achlioptas et al. is that they require acomparison between two unrelated edges which may liein completely distinct regions of the network. Unfortu-nately, since in practice random network models are oftenused to describe decentralized processes (e.g., growth ofsocial or financial networks, spread of disease, etc.), sucha comparison between unrelated edges can be a bit ar-tificial in these circumstances. However, using our cri-terion above we can readily construct variant models inwhich the edges being compared have the feature thatthey share common nodes (which is often the case insocial networks). As an example, consider a model inwhich one simply picks three nodes at random in thenetwork and chooses the lightest edge between them. Acalculation shows that p = 2 for this model and hence it is expected to be explosive (as we have numericallyverified). Another interesting model which can arise insocial networks is the following: An edge between tworandomly selected nodes is proposed. This edge will onlybe “accepted” if both nodes agree to the proposed union.The nodes make their decision as follows: Each node inthe edge picks a second node at random and comparesthe weight of that second node with that of its originalproposed partner. If the original proposed partner nodehas a smaller weight than the second node then the nodeaccepts the edge. If both nodes in the proposed edge ac-cept it then the edge is added, otherwise it is not andthe process is repeated. A straightforward computationshows that this model has p = 2 and thus is expected toexhibit an explosive phase transition (which we have ver-ified numerically). Many other such variations are alsopossible, and can be readily classified using our analysis.For example, if one were to modify the previous model byhaving only one of the nodes in the proposed edge do acomparison, then a quick check would reveal that p = 0,leading one to predict that the transition will not be ex-plosive (as has been numerically verified). Likewise, onecan show that most models in which the heaviest edge ischosen will not exhibit explosive transitions.As a final observation, we note that in this paper wehave identified the underlying mechanism responsible forexplosive transitions in the ER-type networks studied byAchlioptas et al. and provided a criterion that appearsto offer direct guidance for easily recognizing and predict-ing whether or not a given random network will displayan explosive transition. This in turn may prove helpfulin designing new random ER-type network models withdesired characteristics, as in some situations explosivetransitions are desired and in others not. Interestingly,recent observations of explosive transitions have been re-ported by Ziff [15] for two-dimensional percolation mod-els, and by Cho et al. [16] and Radicchi and Fortunato[17] for scale-free networks. It is a significant but cur-rently open question as to how the underlying mechanismand methodology for understanding explosive transitionsin ER-type models is related to the analogous transitionsrecently seen in these other systems.The authors thank Seth Marvel, Joel Nishimura andSteve Strogatz for helpful conversations. EJF’s researchhas been supported in part by the NSF under grantsITR-0325453 and CDI-0835706. [1] R. Albert and A. Barabasi, Reviews of modern physics , 47 (2002).[2] S. Dorogovtsev and J. Mendes, Advances in Physics ,1079 (2002).[3] R. Cohen and S. 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