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Investigating the nature of explosive percolation transition

The Laboratory of Computational Physics is actively involved in the field of investigating the phase transition of various natural and artificial systems. Currently, much effort is being concentrated on the definition of the type of phase transition for a new competitive model named “explosive” percolation: when filling sequentially an empty lattice with occupied sites, instead of randomly occupying a site or bond (according to the classical paradigm), we choose two candidates and investigate which one of them leads to the smaller clustering. The one that does this is kept as a new occupied site on the lattice while the second one is discarded (Figure 1). This procedure considerably slows down the emergence of the giant component, which is now formed abruptly, thus the term “explosive”.

Figure 1: Achlioptas Process according to the sum rule (APSR) for site percolation. White cells correspond to unoccupied sites while colored cells correspond to occupied sites. Different colors (red,green,gray,blue) indicate different clusters. (a) We randomly select two trial unoccupied sites (yellow), noted by A and B, one at a time. We evaluate the size of the clusters that are formed and contain sites A and B, $$s_A$$ and $$s_B$$ respectively. In this example $$s_A = 10$$ and $$s_B = 14$$. (b) According to the Achlioptas Process, we keep site A which leads to the smaller cluster and discard site B.

Following the first publication of Achlioptas et al., a debate was initiated between various teams whether the procedure is continuous or discontinuous. Contributing to these considerations, we have investigated explosive site percolation, both using the product and sum rules. It was found that the exponent $$\beta / \nu$$ is vanishing small for both cases, pointing towards the continuity of the transition. Also, we performed numerical analysis for the case of a reverse Achlioptas process (Figure 2). It was shown that for finite systems there is a hysteresis loop between the reverse and forward procedure (Figure 2). This loop vanishes at infinity, giving strong evidence for the continuity of the “explosive” site percolation (Figure 3). Moreover, “explosive” site and bond percolation seem to belong to a different universality class

 Figure 2: Reverse Achlioptas Process (AP1) for site percolation according to the sum rule. Blue is for the occupied sites while white for the unoccupied sites. Initially, the lattice is fully occupied. (a) An instance of the process. We randomly choose two trial sites (yellow), noted as A and B, and remove them from the lattice. (b) The clusters formed after the removal. (c) We place site A again in the lattice and calculate the size of the cluster in which it belongs, $$s_A = 16$$. (d) We do the same as before for the case of site B and calculate $$s_B = 26$$. We remove site A which leads to the formation of the smaller cluster and keep site B.

Figure 3: (a) Hysteresis loop between a reverse (red dots) and the forward (black squares) Achlioptas process for a $$700\times 700$$ system. (b) The loop vanishes in the thermodynamic limit

Simulations were performed on the EGI. A diagram of the number of jobs and CPU hours consumed per month is shown in Figure 4. We have used extensively the gLite parametric job submission mechanism, using as parameter the different realizations of the system. On average, more than 1000 jobs per simulation were submitted for each lattice size. Considering a typical $$1000 \times 1000$$ lattice, the average time consumed for one run approached 172 minutes. If we had to perform the calculations on a single CPU, this would mean that it would take us 120 days to get complete results for just one lattice size. Using the EGI thus has helped us minimize this time approximately to 172 minutes. This translates to a time gain of the order of $$10^3$$. Moreover, given the availability of more resources, this gain may be even higher. This is a very important feature, because we can numerically analyze systems of the order of $$10^6$$ in a tolerable amount of time.

Figure 4: Number of jobs and CPU hours per month consumed for the simulations

References:

1. D.Achlioptas, R.M. D’Souza and J. Spencer, Explosive Percolation in Random Networks, Science 323,p. 1453 ,(2009)
2. R.A. da Costa, S.N.Dorogovtsev, A.V.Goltsev, and J.F.F.Mendes, “Explosive Percolation” Transition is Actually Continuous, Physical Review Letters 105(25),255701,(2010)
3. P.Grassberger, C. Christensen, G. Bizhani, S-W Son, and M. Paczuski, Explosive Percolation is Continuous, but with Unusual Finite Size Behavior, Phys Rev Lett. 106(22), (2011)
4. O. Riordan and L. Warnke, Explosive percolation is continuous, Science 333 (2011)
5. R.M. Ziff, Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks, Physical Review Letters 103(4),45701,(2009)
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7. N.A.M. Araújo and H.J.Hermann, Explosive Percolation via Control of the Largest Cluster, Physical Review Letters 105(3),035701,(2010)