Variation of the critical percolation threshold in the Achlioptas processes
VVariation of the critical percolation threshold in the Achlioptasprocesses
Paraskevas Giazitzidis, Isak Avramov, and Panos Argyrakis Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece Institute for Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria (Dated: October 17, 2018)
Abstract
We investigate variations of the well-known Achlioptas percolation problem, which uses themethod of probing sites when building up a lattice system, or probing links when building anetwork, ultimately resulting in the delay of the appearance of the critical behavior. In the firstvariation we use two-dimensional lattices, and we apply reverse rules of the Achlioptas model, thusresulting in a speed-up rather than delay of criticality. In a second variation we apply an attractive(and repulsive) rule when building up the lattice, so that newly added sites are either attractedor repelled by the already existing clusters. All these variations result in different values of thepercolation threshold, which are herewith reported. Finally, we find that all new models belong tothe same universality class as classical percolation. a r X i v : . [ phy s i c s . c o m p - ph ] N ov . INTRODUCTION The percolation phase transition [1] has traditionally attracted the interest not only ofphysicists but of scientists in practically all fields over the last five decades due to the fact thatit is a paradigmatic continuous phase transition. More recently, new models have appeared[2] which vary the way by which a lattice is filled up, thus producing a different criticalthreshold value, and very different characteristics of the phase transition, including thequestion if the well known transition is continuous or discontinuous. The delay of criticalityin the percolation problem introduced in [2] has certainly attracted considerable interest inrecent years because it gives one a method to vary the exact location of the critical point,almost at will, by appropriately handling the conditions by which the system is being built upor prepared [3–5]. The initial idea to the direction of discontinuous phase transition includeda 2-particle probe method, which caused the critical point to be considerably delayed. Thus,for a square 2D lattice the critical point moved from p c = 0 .
593 to p c = 0 . p (cid:48) c < p classicalc ). To do this we use the same idea asthe original Achlioptas probe method, but instead of choosing the site that results in thesmaller product or sum of the joining cluster we now choose exactly the opposite, i.e. wechoose to keep the probe site which results in the largest cluster and we discard the otherprobe site. Obviously, this will result in a speed-up of the critical point, i.e. the largestpercolating cluster will now appear earlier than the conventional case. Indeed we find thatinstead of p c = 0 .
593 we now have p c = 0 . p c = 0 . p c = 0 . p c = 0 . II. MODEL DESCRIPTION
We investigate four variations of the well known Achlioptas processes [2]. In the originalmodel one fills the system (lattice or network) by probing at random two candidate sites(or nodes) to be occupied. The one that minimizes the product of the sizes of the clustersthat this site is about to connect is maintained, while the other one is removed. The detailsfor candidate sites maybe different according to the system used. For example, in sitepercolation in a two-dimensional (2D) square lattice there is a maximum of four possibleclusters that can be merged, while in the original Achlioptas processes the newly added bondmay connect only two clusters to form a larger one. However, such details do not affect theoverall system behavior. We have reproduced the data for site percolation product-rule andsum-rule and our results are in excellent agreement with previous publications[8, 9]. Herewe extend both these models by promoting the creation of larger clusters, instead of smallerones. In order to do this we first choose two candidate sites. We calculate the product of thesizes of the clusters that are to be merged for each candidate site separately. Then we keepthe site with the larger of the two products, while we discard the other one. This resultsin the critical point appearing earlier than in normal percolation. Thus, in addition to thedelay of criticality that was suggested by the Achlioptas models, one may now speed-up theappearance of the critical point. 3urthermore, we now introduce two new models, which are based on filling the lattice byprobing the local environment of the site to be added, one using an attractive algorithm,and one using a repulsive one. In the attraction model we start initially with an empty 2Dsquare lattice of linear size L = 1000. We start by probing one site of the lattice at randomand we occupy it only if this site has at least one neighbor occupied. We consider that eachsite has four nearest neighbors. If the chosen site has no nearest neighbors occupied, we thenchoose at random another site of the lattice without investigating if there are any occupiedneighbors (random site percolation) and we occupy it. We continue by probing a secondsite, and so on, and we repeat the same process as previously. In the repulsion model westart again with an empty 2D square lattice of the same linear size L = 1000 and we probea site at random. We then investigate the four neighbor sites of the candidate site and weoccupy it only if there are no occupied neighbors at all. On the other hand, if there is atleast one neighbor site that is occupied, we choose another site at random and we occupyit. For both the attraction model and repulsion model we continue until the lattice is fullyoccupied.These two models, the attraction and the repulsion model, are expected to significantlychange the location of the critical point (as discussed in the section of Results). The locationof the critical point can conceivably be further changed if one is increasing the number ofattempts, for both the attraction and the repulsion models, which is expected to increase thelevel of speed-up or delay of the critical point, respectively. To investigate this we considerthe same 2D square lattice of linear size L = 1000, but this time we introduce a parameter k which gives the number of attempts when probing the lattice sites. Thus, for k = 0 thiscorresponds to conventional random site percolation. k = 1 gives the model explained inthe previous paragraph, where we have only one attempt to search for occupied nearestneighbors before we probe a site of the lattice at random. k = 2 results to two independentattempts. More specifically, in the attraction model we start again with an empty latticeand we probe at random a site to be occupied. If this chosen site has no nearest neighborsoccupied, we probe another site at random and we investigate again for occupied nearestneighbors. That was the second attempt ( k = 2). If again no nearest neighbors are occupiedwe occupy a random site in the lattice without checking its neighbors. It is important tomention that if we find a nearest neighbor occupied during the attempts, the process stopsand a new Monte Carlo step starts from the beginning. This procedure can be extended to4arger k values. We report simulations with different k values for both the attraction andthe repulsion models. III. RESULTS
We monitor the percolation strength P max , which gives the probability of a given occupiedsite to belong to the largest percolating cluster. P max is in the range 0 < P max < P max as a function of the density of occupied sites p . P max = S pL (1)where S is the size (number of sites) of the largest cluster of the system at density p ,and L is the total number of lattice sites.In Fig. 1 we give P max for seven different models. The classical site percolation gives p c = 0 .
593 (full diamonds). The Achlioptas product (PR) and sum-rule (SR) for the delayof criticality are shown with p c = 0 .
755 (open circles) for PR, p c = 0 .
694 (open triangles) forSR. The equivalent Achlioptas processes for the early emergence of criticality now produce p c = 0 .
531 (full circles) for PR, p c = 0 .
543 (full triangles) for SR. The critical thresholdfor the attraction model is p c = 0 .
562 (red full squares) and for the repulsive model is p c = 0 .
610 (red empty squares). While the values of these critical points can be deducedfrom Fig. 1 as the mid-point of the sudden transition, it is usually better for higher accuracyto calculate the first derivative of P max , as this shows the transition in a sharp fashion. We,thus, calculate dP max dp for all seven cases, and we show the results in Fig. 2. We observe thatthe peak of the curvature of the first derivative gives the critical density p c for each model.The values of p c for product-rule and sum-rule as well as for the classical percolation are inexcellent agreement with previous publications [8–10], while values for critical densities forthe reverse processes of product and sum-rule are now calculated here.The Table illustrates the critical threshold for all seven different models that are includedin Fig. 1. To see if these two new processes belong to the same universality class withclassical random percolation, we now calculate the universal critical exponent which doesnot depend on the structural details (topology) of the lattice or on the type of percolation(site, bond). This exponent is the fractal dimension d f . The universality property is a5IG. 1: Plot of percolation strength ( P max ) as a function of the density of occupied sites p for a 2D square lattice of linear size L = 1000. Classical site percolation is indicated withblack full diamonds. The attraction model( k = 1) is indicated with red full squares andthe repulsion model( k = 1) with the empty ones. Sum-rule is indicated with blue triangles.Empty ones are for the delay of criticality, while the full triangles are for the speed-upversion. Product-rule is indicated with green circles. Empty circles are for original productrule for the delay of criticality, while full circles are again for the speed-up version. Thelines are optical guides.general property of second order phase transitions, where the order parameter (here is thesize of the infinite cluster S ) introduces an abrupt increase at the region near the criticalpoint. Eq. 2 shows the logarithmic relation of S to the linear size of the lattice L aroundthe critical point ( p ≈ p c ). S ( L ) ( p ≈ p c ) ∼ L d f (2)Fig. 3 illustrates the scaling of the size of the largest cluster at the critical point S ( p = p c )as a function of L . The slope of the straight line gives the fractal dimension d f , which for theclassical random percolation is well-known and its value has been calculated ( d f ≈ . P max dp ) of percolation strength as a function of thedensity of occupied sites p for a 2D square lattice of linear size L = 1000. Classical sitepercolation is indicated with black full diamonds. The attraction model( k = 1) is indicatedwith red full squares and the repulsion model( k = 1) with the empty ones. Sum-rule isindicated with blue triangles. Empty ones are for the delay of criticality, while the fulltriangles are for the speed-up version. Product-rule is indicated with green circles. Emptycircles are for original product rule for the delay of criticality, while full circles are againfor the speed-up version. The lines are optical guides.[1]. We observe that the slopes of the three straight lines in Fig. 3 have the same value,meaning that all different models examined here belong to the same universality class.One can see an interesting observation in Fig. 1. This is the point that the curves for theattraction model and repulsion model are not symmetric around the curve for the classicalpercolation. This is due to the fact that the rule that we use is not exactly equivalent forboth models. For very low densities ( p < .
1) the majority of the newly added sites arerandomly distributed in the system for both models. There are many isolated sites and themajority of the system consists of empty space. In the attraction model the newly chosensite is isolated in most of the times and thus, a new site is chosen to be occupied (random7ABLE I: The critical threshold p c and the fractal dimension critical exponent d f for allmodels. The results are for site percolation transition on a 2D square lattice. Model p c d f Classical percolation 0.5927 1 . ± . k = 1) 0.5618 1 . ± . k = 1) 0.6100 1 . ± . . ± . . ± . . ± . . ± . percolation). In the repulsion model again we chose at random to occupy a site. Since itis more probable that there are no occupied nearest neighbors, we occupy this specific site,which at these low values of p , occurs most frequently in almost every Monte Carlo step. Butagain, this procedure is equivalent to random percolation. At higher density values but stilllower than the critical, (0 . < p < . p , there is higher probabilityof having random percolation in repulsion model than in the attraction one. In higherdensities (0 . < p <
1) the system undergoes a phase transition and the majority of sites areoccupied. At this last state there are only few sites that are not occupied and the numberof those which are isolated (no nearest neighbors) becomes even smaller. Thus, almost eachnew randomly chosen site is directly occupied in the attraction model because newly added8IG. 3: Finite size scaling of the size of the largest cluster at the critical point S ( L ) ( p = p c ) as a function of the linear lattice size L for the attraction (red circles) and the repulsion(blue triangles) models in log-log coordinates. Black squares are for the classicalpercolation case. The slopes of the straight lines give the fractal dimension d f . All curveshave the same d f , meaning that all different models belong to the same universality class.sites are attracted by occupied neighbors. But in the repulsion model, since the number ofisolated sites is almost zero, random percolation occurs again (because the probability tochoose an isolated site is very small). Thus, at this point, repulsion and attraction modelsare both equivalent to the well-known random percolation. The simulations (Fig. 1) andthe qualitative approach (Fig. 4) show that the rule for the attraction model and the onefor the repulsion are not exactly equivalent during the percolation transition process. Thisis the reason why the curves for these two models are not symmetric to the one of classicalrandom percolation. This asymmetry still exists even for k values higher than one (Fig. 5).We obtain results with different k values for the attraction and repulsion models. Weillustrate the results for both models in the same plot, in Fig. 5 which shows the critical9 a) (b) FIG. 4: The difference in the construction process between attraction model ( a ) andrepulsion model ( b ). White sites are empty, black sites are occupied, while red ones arethose with at least one occupied nearest neighbor. There are more red sites for a fixednumber of black at the repulsion model.threshold p c for both attraction and repulsion models and for different values of attempts k . We observe in Fig. 5 that the asymmetry between the two models still exists even forlarger k values. In addition, we observe that there is a minimum p c for the attraction modelfor k ≈
15. For k >
15 the speed up for the attraction model does not have any furthereffect and further increase of the number of attempts k results again to the delay of thecritical point. The shape of the curve in Fig. 5 with the minimum, giving the delay is quiteunexpected, but it can be explained by the fact that after this minimum value the newlyadded sites are attracted to already existing clusters, with hardly any new clusters beingformed. No new clusters appear (or the rate of appearance is limited). This is similar tocrystallization starting from active centers. Thus, the formation of the infinite cluster whichis spanning the entire system is delayed because there is a small number of clusters which aregrowing simultaneously and need more time to grow in all directions until they touch eachother to form a larger one. In the limit of k → ∞ there is only one nucleus that grows. Thelarger the k value, the stronger is the similarity of the model to that of overall crystallizationkinetics [11]. When a large number of attempts k is made most of the newly deposited sitesare attached to the already existing clusters (except for the initial stages). In a sense at thisrange the filling of the lattice is equivalent to a crystallization process which proceeds froman initially small number of active centers. 10IG. 5: The percolation threshold of sites p c vs the number of independent attempts k forattraction and repulsion models. The attraction model is illustrated with empty squareswhile the repulsion model with full circles. The lines are only optical guides. The bluedashed line indicates p c for classical percolation. In the inset: Critical density of sites p c vsnumber of attempts k to indicate the minimum and the maximum of the two curves. IV. DISCUSSION AND CONCLUSIONS
In this work we have simulated two different models of the percolation phase transition,which depend on the method used to fill the lattice sites. The impetus for this work hasbeen the fact that one can vary the location of the critical point almost at will by varyingthe values of the building parameters of the clusters. This variation alone may be importantwhen preparing a new system with custom-made properties, which can now be tailor-madeaccording to the needed specifications. We simulated the direct and reverse cases of thewell-known Achlioptas processes using both the product and the sum rule. The reverseAchlioptas process is the one in which the probe site maximizes the product (or the sum)of the sizes of the clusters that is about to connect, as opposed to minimizing these as inthe original Achlioptas processes. We introduce two new models based in the control of the11ocal environment of the site to be added. These new models are based on the occupancy ofthe nearest neighbors of the probing sites. The attraction model promotes the merging ofsites to form larger clusters and the repulsion model, which is the reverse process, promotesthe isolation of occupied sites. We located the exact position of the critical density forall seven models that we have examined. Note that the position of the critical point isthe value of density p at the inflection point of the curves given in Fig. 1 or, in a betterway, the density of the peak of the first derivative of percolation strength dP max dp , as givenin Fig. 2. We compared the new findings with well known results, and we found excellentagreement. Our results show that the two new models belong to the same universality classas the classical percolation transition. Also, we extended the newly introduced models byfurther promoting the merging or the isolating of clusters using a parameter which controlsrandom percolation at attraction and repulsion models. We explained the asymmetry thatappears in the attraction and repulsion model around the normal critical transition point.Our results make it possible to control the location of the critical point by controlling themethod of the preparation of the system. In principle, one could use fractional, non-integervalues of k (as done in Ref. [6]) and, thus, produce a new system with the exactly desiredcritical transition point. ACKNOWLEDGMENTS
We thank S. Havlin for useful suggestions and discussions. This work used the EuropeanGrid Infrastructure(EGI) through the National Grid Infrastructures NGI GRNET, Hellas-GRID as part of the SEE Virtual Organization and was supported by the EC-funded FP7project MULTIPLEX Grand number 317532. [1] D. Stauffer and A. Aharony,
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