ARTICLE IN PRESS Physica A 374 (2007) 835–842 www.elsevier.com/locate/physa Microscopic Abrams–Strogatz model of language competition Dietrich Stauffer1,, Xavier Castelló, Vı́ctor M. Eguı́luz, Maxi San Miguel IMEDEA (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain Received 15 March 2006 Available online 14 August 2006 Abstract The differential equation of Abrams and Strogatz for the competition between two languages is compared with agentbased Monte Carlo simulations for fully connected networks as well as for lattices in one, two and three dimensions, with up to 109 agents. In the case of socially equivalent languages, agent-based models and a mean-field approximation give grossly different results. r 2006 Elsevier B.V. All rights reserved. Keywords: Monte Carlo; Language competition 1. Introduction Language competition and extinction is being considered from the point of view of complex systems. Language competition studies the dynamics of language use due to social interactions. It is known that most of the 6000 languages spoken today are in danger, with around 50% of them facing extinction in the current century. Perhaps more important is the distribution of speakers, with 4% of languages accounting for 96% of people and 25% having fewer than 1000 speakers [1]. Many computer simulations of the competition between different languages have appeared, mostly in physics journals, since the publication in 2003 of a model by Abrams–Strogatz [2] for the competition between two languages. Some of them use a mean-field approximation [3–6], while others implement more realistic agent-based models for many [7–10] or few languages [11,12]. A more complete review is given in Ref. [13], and a shorter one in Ref. [14]. Other studies address learning processes of a language [15,16], a question that we do not take into account here. Our main goal in this work is to check to what extent the results of the mean-field approximation of the Abrams–Strogatz model are confirmed by agent-based simulations with many individuals. For the social interaction, we will consider a completely connected network as well as a regular lattice with nearest neighbour interaction in one, two, and three dimensions. Corresponding author. Tel.: +49 221 4705159. E-mail addresses: stauffer@thp.uni-koeln.de (D. Stauffer), xavi@imedea.uib.es (X. Castelló), victor@imedea.uib.es (V.M. Eguı́luz), maxi@imedea.uib.es (M. San Miguel). 1 Visiting from Institute for Theoretical Physics, Cologne University, D-50923 Köln, Euroland, Germany. 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.07.036 ARTICLE IN PRESS 836 D. Stauffer et al. / Physica A 374 (2007) 835–842 2. The Abrams–Strogatz model The model of Abrams–Strogatz studies the competition between two languages, X and Y , in a given society. An individual changes her/his language from Y to X , taking into account: (1) the total number of people speaking this language; (2) its perceived status, a parameter that reflects the attractiveness of a language: access to culture, personal and professional development . . . : The dynamics is as follows: dx=dt ¼ ð1  xÞpYX  xpXY , (1) where the probability pYX to switch from language Y to language X , and the probability pXY for the inverse switch, are given by pYX ¼ sxa ; pXY ¼ ð1  sÞð1  xÞa . (2) Here x is the proportion of people speaking language X , and s its perceived status; 1  x is the proportion of people speaking Y and 1  s its corresponding status. From now on, we will use the word prestige for Abrams–Strogatz status, following common linguistics terminology. The resulting Abrams–Strogatz differential equation for the competition of a language X with prestige s against another language Y with prestige 1  s is dx=dt ¼ ð1  xÞxðxa1 s  ð1  xÞa1 ð1  sÞÞ. (3) Prestige is a parameter in the range 0osp1. The case so12, models the situation of a language with lower prestige X , competing against a more prestigious language Y . Fitting data for several endangered languages, a ’ 1:3 was obtained in Ref. [2]. This equation has three fixed points for aa1: for a41: x ¼ 0 and 1 are stable, and a third one 0ox o1 is unstable. For ao1, stable fixed points become unstable, and vice versa. We have considered a situation where both languages have initially the same number of speakers, xðt ¼ 0Þ ¼ 12. Fig. 1 shows exponential decay for a ¼ 1:31 as well as for the simpler linear case a ¼ 1. From now on we use a ¼ 1. This choice simplifies Eq. (3) into Eq. (4), similar to the logistic equation which was applied to languages before, as reviewed by Wang et al. [17]. dx=dt ¼ ð2s  1Þð1  xÞx. (4) Differential equation: a = 1.31 (+, left line) and 1 (x, right line) for s = 0.1 (+,x) and 0.4 (two lines) 1 0.1 0.01 0.001 fraction 1e-04 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 0 10 20 30 40 50 60 70 80 90 100 time Fig. 1. Fraction of X speakers from Abrams–Strogatz differential equation with a ¼ 1:31 and 1, at status s ¼ 0:1 (heavy symbols at left) and s ¼ 0:4 (two lines at right). For a ¼ 1:31 the decay is faster than for a ¼ 1. ARTICLE IN PRESS 837 D. Stauffer et al. / Physica A 374 (2007) 835–842 We have now two fixed points. For long times and so12, the fraction of speakers x has an exponential decay e . For s ¼ 12 any value of x is a marginally stable stationary solution. ð2s1Þt 3. Agent-based models Differential equation (4) is a mean-field approximation, ignoring the fate of individuals and the resulting fluctuations. To take into account a discrete society, we build an agent-based model with N individuals which in a completely connected network feel the influence of all individuals, while on the d-dimensional lattice they feel only the influence of their 2d nearest neighbours. For lattices therefore, in the probabilities pYX to switch from language Y to language X , and pXY for the inverse switch, x is no longer the global fraction of speakers of language X , but a local density: fraction of X speakers within the 2d nearest neighbours. Initially each person speaks one of the two languages with equal probability: xðt ¼ 0Þ ¼ 0:5. We have considered two different asynchronous updatings: (1) regular updating: the state of every node is updated going through them in an indexed order. (2) Random updating: at each iteration, we choose one agent i at random, and change its language according to the probabilities mentioned above. This is more realistic, but takes more time. In both cases, a time step is defined as N iterations, with every node updated once on average. We find that our results agree qualitatively for both updatings. Our results in the case of nonequivalent languages so12 are shown in Fig. 2 for the fully connected case and in Fig. 3 for the square lattice. Results are qualitatively similar for the completely connected network and the square lattice, as well as for the original differential equation, giving a fast exponential decay of the less prestigious language, until it faces extinction. 4. Socially equivalent languages We consider now the symmetric case s ¼ 12 of competition between two socially equivalent languages. The mean-field approximation fails grossly in this case: the differential equation has x staying at 12 for all times, Fully connected, a = 1, N = 1000(+), 1000,000(x), 1000,000,000(*); differential equation (line) 1 0.1 0.01 fraction 0.001 1e-04 1e-05 1e-06 1e-07 1e-08 0 10 20 30 40 50 60 70 80 90 100 time Fig. 2. Fully connected model with 103 , 106 , 109 agents at s ¼ 0:4 compared with differential equation (rightmost line) at s ¼ 0:4. The three left lines correspond to s ¼ 0:1; 0:2; 0:3 from left to right for N ¼ 109 . The thick horizontal line corresponds to s ¼ 0:5 and N ¼ 106 and changes away from 12 only for much longer times. Figs. 2 and 3 use one sample only and thus indicate self-averaging: the fluctuations decrease for increasing population. In this figure, as well as in Fig. 3, results are obtained with regular updating. ARTICLE IN PRESS 838 D. Stauffer et al. / Physica A 374 (2007) 835–842 Square lattice, L = 101 (+), 301 (x), 1001(*), 3001 (open sq.), 10001 (full sq.); s=.4 (lines: .3, .2, .1) 1 0.1 0.01 fraction 0.001 1e-04 1e-05 1e-06 1e-07 1e-08 0 10 20 30 40 50 60 70 80 90 100 time Fig. 3. L  L square lattice with L ¼ 101 to 10,001 at s ¼ 0:4. The three left lines correspond to s ¼ 0:1; 0:2; 0:3 from left to right for L ¼ 10; 001. The thick horizontal line corresponds to s ¼ 0:5; L ¼ 10; 001 and might deviate only for t4107 . 2-d lattice. Random updating for L = 20, 30, 50, 70, 100 (sq). Regular updating for L = 30, 100 (*) <ε> 100 10-1 10-2 102 104 time Fig. 4. Decay of unstable symmetric solution x ¼ 12 for s ¼ 12 for square lattices of various sizes, with system size increasing from left to right. A semilogarithmic plot, not shown, indicates a simple exponential decay. Simulations shown are done with random updating (straight lines). Some system sizes are also represented for regular updating for comparison (dashed lines). Average over 100 samples. while random fluctuation for finite population systems destabilize this situation and let one of the two languages win over the other, with x going to zero or unity with equal probability. In this symmetric situation, our lattice model becomes similar to the voter model [18]. ARTICLE IN PRESS 839 D. Stauffer et al. / Physica A 374 (2007) 835–842 The symmetric case in a regular lattice can be described in a unified way by looking at the number of lattice neighbours speaking a language different from the centre site. It corresponds to an energy, , in the Ising magnet and measures microscopic interfaces. Initially this number equals d on average. This magnitude is related to the averaged interface density hri, used previously in Ref. [18] to analyse the voter model: hi ¼ 2dhri. (5) 1-d lattice. Random updating for L = 100, 200, 400, 1000, 5000 (sq). Regular updating for L = 200, 1000 (*) 100 <ε> 10-1 10-2 10-3 10-4 102 104 time 3-d lattice. Random upating for L = 10, 12, 15, 22, 26 (sq). Regular updating for L = 15, 22 (*) <ε> 100 10-1 101 102 103 104 time Fig. 5. Same as Fig. 4 but in one (top) and three (bottom) dimensions. For one-dimensional lattice, due to larger times for the separation from the power law decay, and self-averaging for large systems, L ¼ 100; 200 are averaged over 1000 samples; L ¼ 400; 1000 over 200 samples; and L ¼ 5000 over 50 (random updating). For regular update we average over 100 samples. For three-dimensional lattice we average over 100 samples. ARTICLE IN PRESS 840 D. Stauffer et al. / Physica A 374 (2007) 835–842 We present here the results for d-dimensional lattices, with d ¼ 1–3. Fig. 4 shows the results for a square lattice. The energy defined above decays to zero, first possibly as a power law of exponent 0.1 (compatible with the decaying obtained for the voter model, 1= ln t), and then exponentially after a time which increases with increasing lattice size. The first decay describes a coarsening phenomenon, while the exponential decay is triggered by finite size fluctuations. One- and three-dimensional lattices have also been considered, for a more complete analysis. In Fig. 5 we can observe how in one dimension the initial decay follows a power law, t1=2 , while in three dimensions an initial plateau is reached, and thus no coarsening process occurs. This is followed after a time increasing with size by an exponential decay in d ¼ 1; 3 as in two dimensions. Results for both updatings are not quantitatively identical, but give the same qualitative behaviour, including the exponents for the power laws. 100 square lattices with L = 11, 13, 17, 23, 31, 41, 61 from left to right | x - 1/2| 0.1 0.01 1 10 100 (a) 1000 time 100 chains of length L = 101, 201, 501, 1001, 2001, 50001, 10001, 20001 from left to right |x - 1/2| 0.1 0.01 1 (b) 10 100 1000 10000 100000 time Fig. 6. Absolute difference between xðtÞ and xðt ¼ 0Þ ¼ 12 for d ¼ 2 (part a) and 1 (part b, averages for t41000). Average over 100 samples. ARTICLE IN PRESS 841 D. Stauffer et al. / Physica A 374 (2007) 835–842 Fig. 6 shows that the average of jxðtÞ  12j increases in two dimensions roughly as the square-root of time until it saturates at 12, indicating random walk behaviour. (Note that first averaging over x and then taking the absolute value jhxi  12j would not give appropriate results since hxi would always be 1/2 apart from fluctuations.) Fig. 7 shows the dependence on system size of the relaxation time for the extinction of a language associated to the exponential decay mentioned above. Regular updating is shown in Fig. 7a and random updating in Time after which fraction x < 0.4 or x > 0.6 when x(t=0)=0.5; d=1 (+), 2(x), 3(*), 100 samples 1e+06 relaxation time 100000 10000 1000 100 10 100 1000 (a) 10000 lattice size N = L 100000 d Dependence on system size of relaxation time; d=1 (sq), d=2 (+), d=3 (*) 106 relaxation time 105 104 103 102 101 (b) 102 103 104 N Fig. 7. Time for the energy  (number of lattice neighbours speaking different language) to reach some constant fraction of its initial value, versus population N ¼ Ld , in one (+), two (x) and three (*) dimensions. Averaged over 100 samples. Part a uses regular updating: we checked when the fraction x, initially 12 leaves the interval ½0:4; 0:6 on its way to zero or one. The straight lines have slope 1 for d ¼ 2; 3, and 2 for d ¼ 1. Part b uses random updating: we checked when the energy reaches a small fraction of its initial value, taken as 2=L, 0:04, 0:6 for d ¼ 1–3. The exponents for the fitted power laws are 1.99 (d ¼ 1, straight line), 1.08 (d ¼ 2 dotted line) and 0.97 (d ¼ 3 dashed line). ARTICLE IN PRESS 842 D. Stauffer et al. / Physica A 374 (2007) 835–842 Fig. 7b. Both figures are quite similar, with scaling laws for the characteristic time which are compatible with the ones obtained for a voter model [18]: t ’ N 2 in d ¼ 1, t ’ N ln N in d ¼ 2, and t ’ N in d ¼ 3, where N ¼ Ld . Comparisons not shown here between this model for s ¼ 12 and voter model, show how prestige, being a factor reducing the maximum probability to switch to 12, introduces a time delay to the whole dynamics compared to the voter model, but keeps all its qualitative behaviour. 5. Conclusions We conclude that agent-based simulations agree qualitatively for nonequivalent languages in the topologies studied. However, the results differ appreciably from the results of the mean-field approach for the symmetric case s ¼ 12 of two socially equivalent languages: while Eqs. (1) and (2) predict x to stay at x ¼ 12, our simulations in Fig. 4 and later show that after a decay time everybody speaks the same language, bringing the other language to extinction. In a fully connected network and in d ¼ 3 the decay is triggered by a finite size fluctuation, while in d ¼ 1; 2 the intrinsic dynamics of the system causes an initial ordering phenomena in which spatial domains of speakers of the same language grow in size. Other aspects that can be introduced in agent-based models of language competition are the presence of bilingual individuals as well as a complex social structure [5,19,20]. Acknowledgement We acknowledge financial support from the MEC (Spain) through project CONOCE2 (FIS2004-00953). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] D. Crystal, Language Death, Cambridge University Press, Cambridge, 2000. D.M. Abrams, S.H. Strogatz, Nature 424 (2003) 900. M. Patriarca, T. Leppänen, Physica A 338 (2004) 296. W.S.Y. Wang, J.W. Minett, Trans. Philol. Soc. 103 (2005) 121. J. Mira, A. Paredes, Europhys. Lett. 69 (2005) 1031. J.P. Pinasco, L. Romanelli, Physica A 361 (2006) 355. C. Schulze, D. Stauffer, Int. J. Mod. Phys. C 16 (2005) 781; C. Schulze, D. Stauffer, Phys. Life Rev. 2 (2005) 89. T. Tes-ileanu, H. Meyer-Ortmanns, Int. J. Mod. Phys. C 17 (2006) 259. D. Stauffer, C. Schulze, F.W.S. Lima, S. Wichmann, S. Solomon, Physica A, in press, doi:10.1016/j.physa.2006.03.045. V.M. de Oliveira, M.A.F. Gomes, I.R. Tsang, Physica A 361 (2006) 361; V.M. de Oliveira, P.R.A. Campos, M.A.F. Gomes, I.R. Tsang, Physica A 368 (2006) 257. K. Kosmidis, J.M. Halley, P. Argyrakis, Physica A 353 (2005) 595; K. Kosmidis, A. Kalampokis, P. Argyrakis, Physica A 366 (2006) 495. V. Schwämmle, Int. J. Mod. Phys. C 16 (2005) 1519; V. Schwämmle, Int. J. Mod. Phys. C 17 (2006) 103. D. Stauffer, S. Moss de Oliveira, P.M.C. de Oliveira, J.S. Sa Martins, Biology, Sociology, Geology by Computational Physicists, Elsevier, Amsterdam, 2006. C. Schulze, D. Stauffer, Comput. Sci. Eng. 8 (2006) 60. M.A. Nowak, N.L. Komarova, P. Niyogi, Nature 417 (2002) 611. A. Baronchelli, M. Felici, E. Caglioti, V. Loreto, L. Steels, e-prints physics/0509075, 0511201 and 0512045 at arXiv.org. W.S.Y. Wang, J. Ke, J.W. Minett, in: C.R. Huang, W. Lenders (Eds.), Computational Linguistics and Beyond, Academica Sinica: Institute of Linguistics, Taipei, 2004 hhttp://www.ee.cuhk.edu.hk/wsywangi. R. Holley, T.M. Liggett, Ann. Probab. 3 (1975) 643; K. Suchecki, V.M. Eguı́luz, M. San Miguel, Phys. Rev. E 72 (2005) 0361362; V.M. Eguı́luz, M. San Miguel, Europhys. Lett. 69 (2005) 228; M. San Miguel, V.M. Eguı́luz, R. Toral, K. Klemm, Comp. Sci. Engin. 7 (2005) 67. J.W. Minett, W. S-Y. Wang, unpublished. X. Castelló, V.M. Eguı́luz, M. San Miguel, communication 10.51, AKSOE meeting 2006. hhttp://www.dpg-physik.de/static/fachlich/ aksoe/i