STABILITY IN THE SOCIAL PERCOLATION MODELS FOR TWO TO FOUR DIMENSIONS

International Journal of Modern Physics C, Vol. 11, No. 2 (2000 287 300 c World Scientific Publishing Company STABILITY IN THE SOCIAL PERCOLATION MODELS FOR TWO TO FOUR DIMENSIONS ZHI-FENG HUANG Institute for Theoretical Physics, Cologne University, 50923 Köln, Germany E-mail : zfh@thp.uni-koeln.de Received 1 December 1999 Revised 12 December 1999 The social percolation model proposed by Solomon et al. as well as its modification are studied in two to four dimensions for the phenomena of self-organized criticality. Stability in the models is obtained and the systems are shown to automatically drift towards the percolation threshold. Keywords: Percolation; Self-Organized Criticality; Stability. 1. Introduction Problems in social systems have attracted great interest in recent years, and concepts and methods of statistical physics have been applied increasingly in order to understand them. Based on microscopic models, some aspects of social dynamic behavior can be simulated. Recently, Solomon et al. 1 proposed a social percolation model 2 for information propagation across a social network and mutual feedbacks. 3,4 In their original model, a message about the quality of a movie spreads along nearest-neighbor links of a lattice, and one time step consists of a spread across the lattice according to the procedure below. After receiving the message, agents located on sites of the lattice decide to go to the movie if the movie quality q is larger than their personal preferences, i.e., q>, and spread the message further. They then increase their preferences by δp. Otherwise, the agents decide not to go and do not spread the message until the end of this time step. They will correspondingly decrease their by δp. After one time step, the movie quality q increases (or decreases by δq if the message can (or cannot propagate from top to bottom of lattice, that is, if percolation (or no percolation occurs. Self-organized criticality towards the percolation threshold was expected in this model. Solomon et al. 1 have simulated two-dimensional (2D square lattices, and the system was found to automatically approach the site percolation threshold p c =0.593, but only for not too long times if both δq and δp are nonzero. After a very long time, an instability occurs after reaching a metastable plateau in q. 287

288 Z.-F. Huang To understand and avoid this instability, we make a modification of this model. In the original model, the agents who did not receive the message (i.e., unvisited sites keep their preferences at the next time. However, the ones not receiving the message also did not go to the movie, so in the modified model their preferences decrease by δp. For simplicity, in the following the original model of Solomon et al. is denoted as Model I, while the modified model is denoted as Model II. 2. Two-Dimensional Simulations The simulations of both models are on L L square lattices with p c = 0.593. Initially the agents preferences are distributed randomly between 0 and 1, and each time step starts with a full line of movie goers on one boundary of length L. A Leath-type algorithm 5 is used and one time step stops once one site of the cluster reaches the bottom line or the spread stops by itself. 2.1. δq = δp = δ During the evolution of Model I 1 for initial <p c, no percolation occurs and the value of q increases at the beginning, until it evolves to a time with the first percolation. Then q increases more slowly with small fluctuations. After q reaches a long metastable plateau, an instability occurs, and q moves fluctuatingly towards unity. However, for Model II stable results are obtained. The value of q increases at an early time as <p c and no percolation occurs, and then after the first percolation (earlier than that of Model I, it evolves to a very stable plateau where q varies between two final values q f and q f δ, and the distribution of shows a single peak near the final q. Note that during the evolution of movie quality q, there are two kinds of mode: Percolation No Percolation No Percolation Percolation, (1 where movie quality changes from q q δ q q + δ, i.e., every procedure in (1 causes q to increase by δ. At late times this mode (1 is just the instability. The other mode is Percolation No Percolation Percolation No Percolation, (2 where movie quality varies between q and q δ and the metastable (or stable plateau is formed. It is interesting to know why these two similar models behave very differently at late times. After a long enough time, it can be obtained from the simulations that in both models most of sites are visited, i.e., receive the message, and go to movie at the percolation time step, while at the time step with no percolation few sites are visited. For Model I, suppose at a late time t, percolation occurs and the movie quality q is larger than preferences of most of sites. At the next time t +1, q has changed to q δ and of most of sites has changed to + δ. Then at this

Stability in the Social Percolation Models for Two to Four Dimensions 289 time step, movie quality is smaller than most of agents preferences, as shown in the histogram of preferences [Fig. 1(a] where L = 101, δq = δp = δ =01, and initially =0.5, and there is no percolation. After this time, movie quality recovers to q, and preferences of most of sites remain + δ since at the nonpercolation time t+1, most of sites are unvisited and keep their preferences in Model I. Then at time t+2, as shown in Fig. 1(b, although q can be larger than preferences +δ of some sites, we still have nonpercolation, due to the fact that in the square lattice q should be larger than the preferences of roughly more than 59.3% sites for percolation to occur. Later at time t+3, movie quality increases to q +δ which is larger than most of agents preferences, as seen in Fig. 1(c, and then percolation occurs. Thus, at late times the above procedure repeats, corresponding to mode (1 and the instability in Model I. For Model II, since all the unvisited sites will lower their preferences by δ, at time t+2 of the above procedure, preferences of most of sites decrease to instead of remaining at +δ as in Model I. Then we have q at this time step, as clearly indicated in Fig. 2(b, which is the histogram of preferences for Model II, and the case of no percolation cannot occur. Thus, we obtain mode (2 in this modified model as well as a very stable plateau for final q. From Fig. 2, one can also find 60 Model I 101*101 δq=δp=01 =0.5 t=1497 (no percolation q=0.728 40 N( 20 20 0.7260 0.7270 0.7280 0.7290 0.7300 0.7310 0.7320 (a Fig. 1. Histograms N( of the personal preferences at late times for 2D Model I, in a square lattice with L = 101, δq = δp = δ =01, and initially =0.5. (a t = 1497; (b t = 1498; (c t = 1499.

290 Z.-F. Huang 60 Model I 101*101 δq=δp=01 =0.5 t=1498 (no percolation q=0.729 40 N( 20 20 0.7260 0.7270 0.7280 0.7290 0.7300 0.7310 0.7320 (b 60 Model I 101*101 δq=δp=01 =0.5 t=1499 (percolation q=0.730 40 N( 20 20 0.7260 0.7270 0.7280 0.7290 0.7300 0.7310 0.7320 (c Fig. 1. (Continued

Stability in the Social Percolation Models for Two to Four Dimensions 291 30 Model II 101*101 δq=δp=01 =0.5 t=1498 (no percolation q=0.531 20 N( 10 10 0.5291 0.5301 0.5311 0.5321 0.5331 0.5341 0.5351 (a 30 Model II 101*101 δq=δp=01 =0.5 t=1499 (percolation q=0.532 20 N( 10 10 0.5291 0.5301 0.5311 0.5321 0.5331 0.5341 0.5351 (b Fig. 2. Histograms N( of the personal preferences at late times for 2D Model II, with the same parameters as Fig. 1. (a t = 1498; (b t = 1499; (c t = 1500.

292 Z.-F. Huang 30 Model II 101*101 δq=δp=01 =0.5 t=1500 (no percolation q=0.531 20 N( 10 10 0.5291 0.5301 0.5311 0.5321 0.5331 0.5341 0.5351 (c Fig. 2. (Continued that the distribution of preferences automatically drifts towards a single peak near the value of q. The above results are based on an initial value that is lower than the percolation threshold p c =0.593. Simulations with >p c have been done, and the results and analysis are very similar to the above, in particular for the late times. Moreover, if a smaller δ is chosen, we can also get similar results with later appearance of instability for Model I or stable plateau for Model II. Although Model II can give a stable final value of q, this value cannot reach the region very close to p c if the initial is chosen far from it, because of the trapping of final q in the very stable plateau soon after the evolution. But when is close to p c, the final q will be much closer. Thus, if we make more simulations and if the final q of one simulation is set as the initial value of the next one, we can obtain a final value very close to the critical point: q f =0.59 ± 2 for L = 101 and 0.592 ± 03 for L = 1001 (averaged over 50 independent samples for δ =01. 2.2. δq δp For nonzero δq δp the results similar to Sec. 2.1 (δq = δp are found, which is different from the cases of higher dimensions as described below. For Model II the simulations are still stable, and the final stable values of q are different with different δq and δp; while for Model I the instability still occurs. If δq δp, the

Stability in the Social Percolation Models for Two to Four Dimensions 293 metastable region is longer and the instability occurs later with smaller δp, and if δq < δp the instability occurs very fast. 3. Three-Dimensional Simulations The situation in the three-dimensional (3D cubic lattice is different, as p c =0.312 and the percolation occurs when movie quality is larger than the preferences of only about 31.2% sites, which is much less than for the 2D lattice. For a percolating cluster to occur the number of sites with q canbesmallerthanthenumberof sites with q<. Thus, results different from a 2D lattice can be obtained as the percolation occurs more easily here. 3.1. δq = δp = δ For the cubic lattice even the original Model I is stable and gives a nonzero final q value when δq = δp = δ, as verified in Fig. 3 where L = 51, δ =01, and =0.5(> p c or0.2(< p c for both Model I and Model II. It is interesting to understand the reason of stability in the three-dimensional Model I compared with the instability in 2D at late times. Figure 4 shows the histograms of the personal preferences at three continuous late times for initial = 0.5, similar to Fig. 1 for the 2D square lattice. The histogram after a percolation 51*51*51 cubic lattice δq=δp=01 Model I =0.5 Model I =0.2 Model II =0.5 Model II =0.2 0.40 q 0.30 0.20 50 100 150 t Fig. 3. Time evolution of movie quality q in 3D cubic lattice with L = 51, δq = δp = δ =01, and initially =0.5 or 0.2, for both Model I and Model II.

294 Z.-F. Huang 800 Model I 51*51*51 δq=δp=01 =0.5 t=4998 (no percolation q=0.405 600 400 N( 200 200 0.4031 0.4041 0.4051 0.4061 0.4071 0.4081 0.4091 (a 800 Model I 51*51*51 δq=δp=01 =0.5 t=4999 (percolation q=0.406 600 400 N( 200 200 0.4031 0.4041 0.4051 0.4061 0.4071 0.4081 0.4091 (b Fig. 4. Histograms N( of the personal preferences at late times for 3D cubic Model I, with the same parameters as Fig. 3 but =0.5. (a t = 4998; (b t = 4999; (c t = 5000.

Stability in the Social Percolation Models for Two to Four Dimensions 295 800 Model I 51*51*51 δq=δp=01 =0.5 t=5000 (no percolation q=0.405 600 400 N( 200 200 0.4031 0.4041 0.4051 0.4061 0.4071 0.4081 0.4091 (c Fig. 4. (Continued time stes presented in Fig. 4(a, from which nonpercolation of the lattice is obtained, just as in Fig. 1(a for the 2D lattice. Thereafter, although the histogram for the 3D lattice [Fig. 4(b] and the previous one for the 2D lattice [Fig. 1(b] look very similar, they correspond to different results: the spanning cluster (percolation in the 3D lattice is obtained from Fig. 4(b since only about 31.2% sites with q> are needed for percolation, while no percolation occurs in Fig. 1(b as p c is much larger there and more sites with q> are needed for percolation. Thus, in the 3D lattice, we have the stable evolution mode (2 at late times, corresponding to the stable simulations and the existence of self-organized phenomena in Model I. The corresponding histograms at late times for Model II are similar to the histograms for the 2D lattice (Fig. 2 and also correspond to stable mode (2. As in Model II the unvisited sites also decrease their and then preferences of the lattice change faster than in Model I, the stable region of the model is reached faster, in particular for initial <p c as shown in Fig. 3. Similar to the case for the 2D lattice, the final q values obtained in these models cannot approach p c if the initial is chosen far from it, and it will be closer with closer. Thus, using the same procedure as the 2D lattice, i.e., the final q of one simulation is set as the initial value of the next one, after some simulations the final q will be constant: q =0.323 ± 02 (Model I and 0.312 ± 05 (Model II, which are obtained by averaging 30 samples for L =51andδ=01. Thus, the automatic drift towards the critical point is shown in particular for Model II. We

296 Z.-F. Huang have simulated Model I in larger lattices and the final q is closer to the critical point with larger lattice: q =0.317 ± 01 for L = 101 and 0.314 for L = 201 (averaged over 30 samples. 3.2. δq δp For 3D cubic lattice, the results of both models are similar to that of Sec. 3.1 if δq > δp, and the final stable nonzero q are different with different values of δq and δp. However, if δq < δp and δq is small enough (still nonzero, the results are different from above in both models. The final q value will decay to zero whether the initial is larger or smaller than p c, as shown in Fig. 5 where L = 51, δq =001, δp =01, and =0.5 or 0.2. Moreover, the movie quality q in Model I decays faster with smaller δq, while in Model II, q decays slower with smaller δq. 3.2.1. Initially >p c In the simulations when >p c percolation occurs from beginning, Fig. 5 (for =0.5 shows that the q value decreases until zero. For Model I, percolation appears most of the time even q is small, since the condition for the appearance of the spanning cluster is easier here in comparison to the 2D lattice, and δq is so small that q can be always larger than preferences of enough sites. But at late times when 0.80 0.60 51*51*51 cubic lattice δq=001 δp=01 Model I =0.5 Model I =0.2 Model II =0.5 Model II =0.2 0.40 q 0.20 0 0.20 200 400 600 800 1000 t Fig. 5. Time evolution of movie quality q in 3D cubic lattice with L = 51, δq =001, δp =01, and initially =0.5or0.2,forbothModelIandModelII.

Stability in the Social Percolation Models for Two to Four Dimensions 297 q decreases to a small enough value, it is possible that nonpercolation can occur at some times and q slightly increases by δq. Then, the continuous percolations recover and the q value decays again. In Model II the preferences of unvisited sites also decrease by δp (much larger than δq, so nonpercolation cannot occur even when q is very small, as found in the simulations, and percolation remains until q decays to zero. In Model I when δq is smaller, the possibility of nonpercolation occurring during the percolation processes is smaller, then q decays faster. On the other hand, in Model II the percolation occurs and q decays continuously without break, so q decays slower with smaller δq, which is in agreement with the results of simulations. 3.2.2. Initially <p c InModelI,asatearlytimeswhenq<p c, no percolation occurs; q then increases slowly with faster changes of, as shown in Fig. 5. When q increases to the value close to p c, percolation occurs and the later procedure is similar to the one described in Sec. 3.2.1. In Model II, although there is no percolation and q increases at a very early time, the spanning cluster occurs soon in the lattice (much faster than Model I, asshowninfig.5for =0.2, since the here of unvisited sites also decreases by δp (much larger than δq and the preferences of the lattice change much faster. The situation of later times is the same as the one described in Sec. 3.2.1. 4. Higher-Dimensional Simulations For the higher dimensions (d >3, hypercubic results similar to the 3D models are expected to obtain, since for d>3 hypercubic lattices p c is smaller and it is easier for percolation to occur. Figure 6 shows the evolution of q value for d = 4 with p c =0.197, where L = 31, δq = δp =01, and =0.5 or 0.1. In the simulations, results similar to 3D are found, which is as expected, i.e., when δq = δp, the simulations of both models are stable and finally reaches a nonzero stable value; when δq < δp and δq is small enough, the final q will decay to zero whether the initial is larger or smaller than p c. All the results above (lattices of d 2 are based on the models where the spread of movie information will stop upon reaching the bottom. The models (I and II with the revision that the information spread continues after reaching the bottom have also been simulated, and similar results have been obtained. 5. Sales Rate When simulating the sales processes of a movie (product, it is interesting to study the time behavior of sales rate which is a main property measured by the marketing people. 6 It has been found that in a real market, the sales rate of a successful product

298 Z.-F. Huang 0.50 0.40 31*31*31*31 hypercubic lattice δq=δp=01 Model I =0.5 Model I =0.1 Model II =0.5 Model II =0.1 q 0.30 0.20 0.10 50 100 150 t Fig. 6. Time evolution of movie quality q in 4D hypercubic lattice with L = 31, δq = δp =01, and initially =0.5or0.1,forbothModelIandModelII. 300 Model II 51*51*52 δ=01 =0.2 start with 1 active point 132 500 1000 5000 200 sales rate 100 5 10 15 time of spread Fig. 7. Sales rate (number of new sites going to movie versus time of message spread to nearest neighbors for 3D Model II, at different simulation steps of percolation in stable plateau with the same q =0.331. Each run starts with 1 active site on the boundary.

Stability in the Social Percolation Models for Two to Four Dimensions 299 first shows an exponential rise, and then decays fast after reaching a peak region with large fluctuations. The sales rate of a movie has been measured in the social models here. Each simulation time sten the above sections corresponds to a sales process of a particular movie. A new unit time stes now introduced and represents the time necessary for a site (agent to spread a message to its nearest neighbors (denoted as time of spread in each simulation step. The sales rate is then presented as the number of new sites deciding to go to the movie at a new time of spread. Figure 7 shows the results in 3D Model II with L = 51, δq = δp = δ = 01, and =0.2, at different simulation steps of percolation (corresponding to successful sales which are all in the stable region of simulations with the same movie quality q =0.331, and starts with only one randomly selected active site of movie goer on the boundary. A time behavior of sales rates similar to that of the real market is obtained in the figure, and the change of the forms of sales rate at different simulation steps is clearly shown, that is, much smoother for later steps. It is interesting to note that at late steps, the sales rate reaches a round-peak distribution form (Gaussian-like, and the plots are smooth and overlap. In the no percolation simulation steps, the sales rate is very small as few sites are excited and the product fails, in particular at late steps where sales rate directly decreases from the starting value to zero in a new time step. 6. Discussions and Conclusions We have found phenomena of self-organized criticality in social percolation models, in particular for Model II in which the personal preferences of the system change faster to match the change in movie quality due to a modification in which agents not receiving the message also decrease their preferences; the system then automatically drifts towards the critical point. In the original Model I, the system is unstable in 2D and self-organized percolation transition exist in higher dimensions, where agents (sites have more links and connections with each other and where perolation occurs more easily with much lower p c. Thus, it is expected that the instability in 2D can disappear if the message is allowed to spread beyond nearest neighbors, which corresponds to more links among sites or agents and a lower percolation threshold (smaller than 0.5 for next nearest-neighbor links, and much smaller for larger neighborhood 2. Then it is easier for percolation to occur, just as in higher dimensions. Consequently, there is no fundamental difference between 2D and higher dimensions for social percolation models, and stability exists for large societies with more connections among agents. Acknowledgments The author thanks D. Stauffer, N. Jan, and S. Solomon for very helpful discussions and comments. This work was supported by SFB 341.

300 Z.-F. Huang References 1. S. Solomon, G. Weisbuch, L. de Arcangelis, N. Jan, and D. Stauffer, Physica A 277, 239 (2000; Z. F. Huang, Eur. Phys. J. B, in press (2000; E. Ahmed and H. A. Abdusalam, Eur. Phys. J. B, in press. 2. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor and Francis, London, 1994; M. Sahimi, Applications of Percolation Theory (Taylor and Francis, London, 1994; A. Bunde and S. Havlin, Fractals and Disordered Systems (Springer, Berlin-Heidelberg, 1996. 3. S. Moss de Oliveira, P. M. C. de Oliveira, and D. Stauffer, Evolution, Money, War and Computers (Teubner, Stuttgart-Leipzig, 1999. 4. F. Schweitzer (ed., Self-Organization of Complex Structures: From Individual to Collective Dynamics (Gordon and Breach, Amsterdam, 1997. 5. H. G. Evertz, J. Stat. Phys. 70, 1075 (1993. 6. The author thanks S. Solomon for introducing him this interesting subject and many very helpful discussions.