The Sandpile Model on Random Apollonian Networks

Transcription

1 1 The Sandpile Model on Random Apollonian Networks Massimo Stella Bak, Teng and Wiesenfel originally proposed a simple model of a system whose dynamics spontaneously drives, and then maintains it, at the edge of stability: a sandpile. The sandpile model was inspired to natural systems, exhibiting robust non-local correlations, in order to capture the main features of their self-organizing criticality (SOC). In this work we briefly review the BTW model, also by numerically implementing it, and then generalize it on scale-free topologies such as Random Apollonian Networks. Our numerical results suggest that the scale-free topology actually preserves the presence of long-range correlations but lowers their robustness. Introduction and Background Bak, Tang and Wiesenfeld developed their seminal ideas behind Self- Organized Criticality in their paper, [Bak et al., 1988], mainly by the means of extensive numerical experiments. In addition, their work has been successfully applied in a variety of fields [Bak, 1996], such as earthquakes, brain activity and solar flames among many others. Self-Organized Criticality is exhibited by dynamical systems, with both spatial and temporal degrees of freedom, that spontaneously evolve towards an unstable steady state, a good example being the sand trickling down in a hourglass. In the lower part of the latter, individual grains fall, remain close to where they land and an initially flat pile starts forming up. As the trickling continues, the pile becomes steeper and steeper. Eventually, its slope reaches a certain threshold (i.e. the angle of repose) and sand slides start happening. As more sand trickles down, slides become bigger and bigger and some of them may even span all or most of the pile, testifying the instability (i.e. criticality ) of the whole system. This example encapsulates the quintessence of Complexity: even though each sand grain moves according to simple laws, the avalanche dynamics of the whole system cannot be trivially understood by analysing the moves of the single grains separately, as in a reductionistic 1 approach. Numerical Implementation We implemented the BTW model in Mathematica. We simulated a 2D cellular automaton (CA), in which each cell (x, y) is in a discrete state s x,y,t at time t, equal to the number of sand grains on (x, y). Sand is added to one random cell at each time step. If a cell contains more than a critical number c = 4 of grains, then a toppling happens and As in an hourglass, dry sand grains trickling down on a sandy beach pile up. Initially they tend to be static because of friction forces. However, if sand keeps falling down, eventually the pile collapses, spontaneusly. Picture from [Bak, 1996]. 1 Remarkably, in [Bak et al., 1988], the authors always use critical to address the complexity of these systems, while in Bak s book [Bak, 1996], published 8 years later, SOC is clearly linked to the abundance of complex systems in nature. This small detail testifies the rapid development of Complexity Science in the Nineties.

2 2 s x,y,t+1 = s x,y,t toppled grain. 4 while each first-neighbor of (x, y) receives one Figure 1: The BTW sandpile model is not a properly realistic model of sand but it can be visualized as a sand dune system (left) where however dunes are discrete piles on a grid (right). The height of each pile in time is the cell state of the cellular automaton. Additionally, the slidings could bring one or more of the cell neighbors above the stability threshold c, so that they undergo through the same toppling procedure. This nonlinear diffusion of unstability constitutes an avalanche, its size s being the number of slidings originated by the initial single perturbation. Constantly adding sand grains to the system drives it towards a critical state, where even a small noise can grow over all length scales, i.e. a local perturbation can lead to anything, from a single sliding to an avalanche spanning through the whole lattice. This peculiar feature expresses the lack of a characteristic length scale, that also implies a lack of a time scale, because of the nonlinearity of the avalanche dynamics, so that we expect for the statistics concerning the avalanches to be scale-free. For instance, if f (s) is the frequency of an avalanche of size s, then in the critical state f (s) has to be invariant under scale trasformations s! as, with a 2 R. A compatible functional form for f (s) is, therefore, a power-law 2, f (s) µ s t. This surmise was confirmed by the numerical experiments in [Bak et al., 1988]. We reproduced the results shown in Figure 3 of [Bak et al., 1988], with the same 2D CA with N = = 2500 cells, with fixed 3 boundary conditions. As in the original simulation, within our Mathematica code we added sand to one randomly chosen cell (x, y) until it had c + 1 grains, then we measured the size of the subsequent avalanche. We registered avalanches, without including in our statistics the initial transient phase and averaging on 10 initial random configurations, contrary to the 200 of the original paper. For each avalanche, our code executed one check on each of the N cells of the CA, until the slidings stopped, the mean avalanche size being estimated as N 0.9, [Garber and Kantz, 2009]. These two factors give rise to a lower t! t A 2D cellular automaton with 5 5 cells. The toppling process affects only the red cell, with more than 4 grains on it. A sliding happens and 4 grains topple down on the (Von Neumann) neighbors, in blue, of the red cell. This avalanche has size 1, i.e. it includes one sliding. 2 Power-laws are, in fact, a solution to the functional equation g(ax) =bg(x), with a, b 2 R. Scale invariance just asks that a scale in the variable x preserves the functional form of g. For an exaustive review of the scalefree property in nature we refer to [Caldarelli, 2007]. 3 As in the original simulations, we fixed the boundaries outside of the CA to always have 0 height in time, i.e. the grains crossing the borders of the 2D grid fall away.

3 3 bounded estimate of O(N 1.9 ) for the time complexity of our code, for each avalanche, in terms of the CA-size N. Figure 2: Log-log plot of avalanche size s vs avalanche frequency f (s) =D(s) in the original model (left) and in our simulated one (right) for the same CA with N = = 2500 cells. As displayed in Figure 2, our numerical results agree with [Bak et al., 1988]. The frequence f (s) of avalanches of size s is a power-law, exhibiting some fluctuations 4 in the central region (s 2 [20, 120]) and a finite size effect on its tail (s > 500). A fitting procedure retrieves a scaling exponent t 2D = (0.92 ± 0.01) statistically significant (t- Statistics ' 139.5, p-value ' ), in reasonable agreement with the original measurement t BTW ' 1 of [Bak et al., 1988]. Extension of the Original Model The sandpile model exhibits strongly interdependent spatial and temporal degrees of freedom, so that its own topological properties actually affect its own dynamics. In this section we want to investigate an extension of the original BTW model on non-regular graphs different to a 2D lattice. In particular, we use Random Apollonian Networks (RANs), planar graphs related to the Apollonian Gasket problem, see [Andrade Jr et al., 2005] as reference. RANs are growing networks. Their starting configuration is a triangle with 3 connected nodes. At each discrete time step t, a triangle face T is chosen uniformly at random, a new vertex is inserted inside and connected to the three vertices delimiting T. In the last few years, RANs became increasingly popular as spatial network null models for real world networks, mainly because they are scale-free, small-worlds and planar, [Andrade Jr et al., 2005, Caldarelli, 2007]. In our extension, each network node plays the role of a cell in the state s x,y,t. Given that in the original model the threshold c is also the degree of the bulk nodes, the natural extension in our case is to define the toppling threshold c i for node i as its degree k i (i.e. the number of other nodes it is connected to). At each time step, as in the original model, we chose at random a node i, dropped sand on it 4 A point, addressed not in the original paper but in [Bak, 1996], is that the noise on f (s) is to be attributed not only to statistical fluctuations but also and mainly to the avalanches reaching the boundaries. In fact, the latters do not have the same simmetry properties of bulk avalanches and their contribution to the statistics vanishes as slowly as N 1/2 [Dhar, 2006]. We ruled those avalanches out of our statistics, to compensate our average on a smaller number of initial configurations. A point only sketched in the original paper are the finite-size cut-offs on f (s). A recent approach, when fitting power-laws, is using a logarithmic data binning [Dhar, 2006, Garber and Kantz, 2009], for a better visualizaton of such finite-size biases. Here we report such plot for a 2d lattice with N equal to 900 (dot-dashed red), 2500 (dashed blue) and 4900 (dotted orange) nodes, for 5000 avalanches, obtained by also using code from [Clauset et al., 2009]. F(s) here is the probability of an avalanche of size equal to or greater than s to happen. A scaling in the cut-off depending on N, i.e. a finite-size effect, is evident.

4 4 until it had k i + 1 sand grains, let it topple and registered the size of the subsequent avalanche. Given the difficulties arising in defining fixed border conditions for a non-regular network, we modified the dynamics of the BTW model, in order to avoid unending avalanches, by randomly depleting a node j of all but one sand grains on it, i.e. by forcing s j,t = 1. Our results for 10 4 avalanches are reported in Figure 3 and they are averaged on 10 different RANs (each with 2500 nodes). Figure 3: An example of RAN after 4 time steps (left). The color and size of nodes express their state s, with bigger and darker nodes having more grains. Results for the frequency f (s) of avalanches of size s. A fitting procedure retrieves a scaling exponent t RAN = (1.41 ± 0.09) statistically significant (t-statistics ' 158.3, p-value ' ), despite the presence of finite-size fluctuations for lower frequencies. Discussion and Conclusions Our implementation of the original BTW sandpile model allows for predictions on the frequency f (s) µ s 0.92 of avalanches of size s that are in rather good agreement with [Bak et al., 1988]. Furthermore, our extended model of sandpiles on RANs still retrieves a power-law behavior for f (s) but with a lower scaling exponent t RAN ' 1.4. Our numerical findings indicate that the scale-free structure sensibly alters the avalanche dynamics. Compared to the ordinated structure of the 2D lattice, the scale-free topology still allows for the system to reach the critical state but it also decreases the average avalanche size, determining a lowering of the scaling exponent t. This phenomenon is to be attributed to the presence of many hubs inside the network, acting as reservoirs and obstacolating the formation of further slidings. To the best of our knowledge, there has not yet been an attempt to theoretically address the role of hierarchical non-regular structures, evidently present in RANs, inside the avalanche dynamics context and this would be an exciting perspective for future works. A logarithmically binned plot for a RAN with N equal to 1000 (dashed red) and 2500 (dot-dashed blue), for 5000 avalanches, obtained by also using code from [Clauset et al., 2009]. F(s) is the probability of an avalanche of size equal to or greater than s to happen. A scaling in the cut-off depending on N, i.e. a finite-size effect, is found also in the case of RANs, without fixed borders.

5 5 References José S Andrade Jr, Hans J Herrmann, Roberto FS Andrade, and Luciano R da Silva. Apollonian networks: Simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. Physical Review Letters, 94(1):018702, Per Bak. How nature works: the science of self-organized criticality. Copernicus, Per Bak, Chao Tang, Kurt Wiesenfeld, et al. Self-organized criticality. Physical review A, 38(1): , G. Caldarelli. Scale-Free Networks: Complex Webs in Nature and Technology. Oxford Finance Series. OUP Oxford, ISBN URL Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman. Powerlaw distributions in empirical data. SIAM Rev., 51(4): , November ISSN Deepak Dhar. Theoretical studies of self-organized criticality. Physica A: Statistical Mechanics and its Applications, 369(1):29 70, A. Garber and H. Kantz. Finite-size effects on the statistics of extreme events in the btw model. The European Physical Journal B - Condensed Matter and Complex Systems, 67(3): , 2009.