Microscopic Abrams-Strogatz model of language competition Dietrich Stau(cid:11)er*, Xavier Castello(cid:19), V(cid:19)(cid:16)ctor M. Egu(cid:19)(cid:16)luz, and Maxi San Miguel IMEDEA (CSIC-UIB), Campus Universitat Illes Balears E-07122 Palma de Mallorca, Spain * Visiting from Institute for Theoretical Physics, Cologne University, D-50923 Ko(cid:127)ln, Euroland e-mail: fxavi,maxi,[email protected], stau(cid:11)[email protected] Abstract: The di(cid:11)erential equation of Abrams and Strogatz for the competition between two languages is compared with agent-based 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 (cid:12)eld approximation give grossly di(cid:11)erent results. 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 inthe current century. Perhaps more importantis 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 di(cid:11)erent lan- guages 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 (cid:12)eld approximation [3, 4, 5, 6], while others implement more realistic agent-based-models for many [7, 8, 9, 10] or few languages [11, 12]. A more complete review is given in [13], and a shorter one in [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 (cid:12)eld approximation of the Abrams-Strogatz model are con(cid:12)rmed by 1 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 1, 2, and 3 dimensions. 2.THE ABRAMS-STROGATZ MODEL The model of Abrams-Strogatz studies the competition between two lan- guages, X and Y, in a given society. An individual changes her/his language fromYtoX,taking intoaccount: 1)the totalnumber ofpeople speaking this language; 2) its perceived status, a parameter that re(cid:13)ects the attractiveness of a language: access to culture, personal and professional development,... The dynamics is as follows: dx=dt = (1(cid:0)x)p (cid:0)xp (1) YX XY where the probability p to switch from language Y to language X, and the YX probability p for the inverse switch, are given by, XY p = sxa; p = (1(cid:0)s)(1(cid:0)x)a (2) YX XY HerexistheproportionofpeoplespeakinglanguageX,andsitsperceived status; 1(cid:0)xistheproportionofpeoplespeakingYand1(cid:0)sitscorresponding status. From now on, we will use the word prestige for Abrams-Strogatz status, following common linguistics terminology. The resulting Abrams-Strogatz di(cid:11)erential equation for the competition of a language X with prestige s against another language Y with prestige 1(cid:0)s is dx=dt = (1(cid:0)x)x xa(cid:0)1s(cid:0)(1(cid:0)x)a(cid:0)1(1(cid:0)s) (3) (cid:16) (cid:17) Prestige is a parameter in the range 0 < s (cid:20) 1. The case s < 1=2, 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 [2]. This equation has three (cid:12)xed points for a 6= 1: For a > 1: x=0 and x=1 (cid:3) are stable, and a third one 0 < x < 1 is unstable. For a < 1, stable (cid:12)xed points become unstable, and vice versa. We have considered a situation where both languages have initially the same number of speakers, x(t = 0) = 1=2. Fig. 1 shows exponential decay for a = 1:31 as well as for the simpler linear case a = 1. From now on we 2 use a = 1. This choice simpli(cid:12)es (3) into equation (4), similar to the logistic equation which was applied to languages before, as reviewed by [17]. dx=dt = (2s(cid:0)1)(1(cid:0)x)x (4) We have now two (cid:12)xed points For long times and s < 1=2, the fraction of speakers x has an exponential decay e(2s(cid:0)1)t. For s = 1=2 any value of x is a marginally stable stationary solution. 3.AGENT BASED MODELS Di(cid:11)erential equation (4) is a mean-(cid:12)eld approximation, ignoring the fate of individuals and the resulting (cid:13)uctuations. To take into account a discrete society, we build an agent-based model with N individuals which in a com- pletely connected network feel the in(cid:13)uence of all individuals, while on the d-dimensional lattice they feel only the in(cid:13)uence of their 2d nearest neigh- bors. For lattices therefore, in the probabilities p to switch from language YX Y to language X, and p for the inverse switch, x is no longer the global XY 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: x(t = 0) = 0:5. We have considered two di(cid:11)erent asynchronous updatings: 1) regular up- dating: 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 de(cid:12)ned as N iterations, with every node updated once on average. We (cid:12)nd that our results agree qualitatively for both updatings. Our results in the case of non equivalent languages s < 1=2 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 thesquarelattice, aswellasfortheoriginaldi(cid:11)erentialequation, givingafast exponential decay of the less prestigious language, until it faces extinction. 4.SOCIALLY EQUIVALENT LANGUAGES We consider now the symmetric case s = 1=2 of competition between two socially equivalent languages. The mean (cid:12)eld approximation fails grossly in 3 this case: the di(cid:11)erential equation has x staying at 1/2 for all times, while random (cid:13)uctuation for (cid:12)nite 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]. The symmetric case in a regular lattice can be described in a uni(cid:12)ed way by looking at the number of lattice neighbours speaking a language di(cid:11)erent from the centre site. It corresponds to an energy, (cid:15), in the Ising magnet and measures microscopic interfaces. Initially this number equals d on average. This magnitude is related to the averaged interface density < (cid:26) >, used previously in [18] to analyze the voter model: < (cid:15) >= 2d < (cid:26) > (5) We present here the results for d-dimensional lattices, with d = 1;2;3. Fig.4 shows the results for a square lattice. The energy de(cid:12)ned above decays to zero, (cid:12)rst possibly as a power law of exponent 0.1 (compatible with the decaying obtained for the voter model, 1=lnt), and then exponentially after atime which increases withincreasing lattice size. The(cid:12)rst decay describes a coarseningphenomenon, whiletheexponentialdecayistriggeredby(cid:12)nitesize (cid:13)uctuations. 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, t(cid:0)1=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 = 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. Fig.6 shows that the average of jx(t)(cid:0)1=2j increases in two dimensions roughlyasthesquare-rootoftimeuntilitsaturatesat1/2,indicatingrandom walk behavior. (Note that (cid:12)rst averaging over x and then taking the absolute value j < x > (cid:0)1=2j would not give appropriate results since < x > would always be 1/2 apart from (cid:13)uctuations.) 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 Fig.7b. Both (cid:12)gures are quite similar, with scaling laws for the characteristic time which are compatible with the ones obtained for a voter model [18]: (cid:28) ’ N2 4 in d = 1, (cid:28) ’ N lnN in d = 2, and (cid:28) ’ N in d = 3, where N = Ld. Comparisons not shown here between this model for s = 1=2 and voter model, show how prestige, being a factor reducing the maximum probability to switch to 1/2, introduces a time delay to the whole dynamics compared to the voter model, but keeps all its qualitative behaviour. CONCLUSIONS We conclude that agent-based simulations agree qualitatively for non equivalent languages in the topologies studied. However, the results di(cid:11)er appreciably from the results of the mean-(cid:12)eld approach for the symmetric case s = 1=2 of two socially equivalent languages: while Eqs.(1,2) predict x to stay at x = 1=2, 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 = 3 the decay is triggered by a (cid:12)nite size (cid:13)uctuation, while in d = 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]. We acknowledge (cid:12)nancial support fromthe MEC (Spain) through project CONOCE2 (FIS2004-00953). References [1] D. Crystal, Language death (Cambridge: CUP, 2000). [2] D.M. Abrams and S.H. Strogatz, Nature 424 (2003) 900. [3] M. Patriarca and T. Leppa(cid:127)nen, Physica A 338 (2004) 296. [4] W.S.Y. Wang and J.W. Minett, Trans. Philological Soc.103 (2005) 121. [5] J.Mira and A. Paredes, Europhys. Lett. 69 (2005) 1031. [6] J.P. Pinasco and L. Romanelli, Physica A 361 (2006) 355. 5 [7] C. Schulze and D. Stau(cid:11)er, Int. J. Mod. Phys. C 16 (2005) 781; Physics of Life Reviews 2 (2005) 89; [8] T. Te(cid:24)sileanu and H. Meyer-Ortmanns, Int. J. Mod. Phys. C 17, No. 3, 2006, in press. [9] D. Stau(cid:11)er, C. Schulze, F.W.S. Lima, S. Wichmann and S. Solomon, e-print physics/0601160 at arXiv.org. [10] V.M. de Oliveira, M.A.F. Gomes and I.R. Tsang, Physica A 361 (2006) 361; V.M. de Oliveira, P.R.A. Campos, M.A.F. Gomes and I.R. Tsang, e-print physics/0510249 at arXiv.org for Physica A. [11] K. Kosmidis, J.M. Halley and P. Argyrakis, Physica A, 353 (2005) 595; K.Kosmidis, A. Kalampokis and P.Argyrakis, physics/0510019 in arXiv.org to be published in Physica A. [12] V. Schwa(cid:127)mmle, Int. J. Mod. Phys. C 16 (2005) 1519; ibidem 17, 2006, in press. [13] D. Stau(cid:11)er, S. Moss de Oliveira, P.M.C. de Oliveira, J.S. Sa Martins, Biology, Sociology, Geology by Computational Physicists, Elsevier, Am- sterdam 2006. [14] C. Schulze and D. Stau(cid:11)er, Comput. Sci. Engin. 8 (2006) in press. [15] M.A. Nowak, N.L. Komarova and P. Niyogi, Nature 417 (2002) 611. [16] A. Baronchelli, M. Felici, E. Caglioti, V. Loreto, L. Steels, e-prints physics/0509075, 0511201 and 0512045 at arXiv.org. [17] W.S.Y. Wang, J. Ke, J.W. Minett, in: Computational linguistics and beyond, eds. C.R. Huang and W. Lenders (Academica Sinica : Institute of Linguistics, Taipei, 2004); www.ee.cuhk.edu.hk/(cid:24)wsywang [18] R. Holley and T.M. Liggett, Ann. Probab. 3 (1975) 643; K. Suchecki, V.M. Egu(cid:19)(cid:16)luz and M. San Miguel, Phys. Rev. E 72 (2005) 0361362 and Europhys. Lett. 69 (2005) 228; M. San Miguel, V.M. Egu(cid:19)(cid:16)luz, R. Toral and K. Klemm, Comp. Sci. Engin. 7 (Nov/Dec 2005) 67. [19] J.W. Minett, W. S-Y. Wang. (unpublished). 6 [20] X. Castello(cid:19), V.M. Egu(cid:19)(cid:16)luz, M. San Miguel; communication 10.51, AK- SOE meeting 2006. http://www.dpg-physik.de/static/fachlich/aksoe/ 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 1e-04 n o cti 1e-05 a fr 1e-06 1e-07 1e-08 1e-09 1e-10 0 10 20 30 40 50 60 70 80 90 100 time Figure 1: Fraction of X speakers from Abrams-Strogatz di(cid:11)erential equation with a = 1:31 and a = 1, at status s = 0:1 (heavy symbols at left) and s = 0:4 (two lines at right). For a = 1:31 the decay is faster than for a = 1. 7 Fully connected, a = 1, N = 1000(+), 1000,000(x), 1000,000,000(*); differential equation (line) 1 0.1 0.01 0.001 fraction 1e-04 1e-05 1e-06 1e-07 1e-08 0 10 20 30 40 50 60 70 80 90 100 time Figure 2: Fully connected model with 103; 106; 109 agents at s = 0:4 com- pared with di(cid:11)erential equation (rightmost line) at s = 0:4. The three left lines correspond to s = 0:1; 0:2; 0:3from leftto right forN = 109. The thick horizontal line corresponds to s = 0:5 and N = 106 and changes away from 1/2 only for much longer times. Figs. 2 and 3 use one sample only and thus indicate self-averaging: The (cid:13)uctuations decrease for increasing population. In this (cid:12)gure, as well as in Fig.3, results are obtained with regular updating. Square lattice, L = 101 (+), 301 (x), 1001(*), 3001 ( open sq.), 10001 (full sq.); s=.4 (lines: .3, .2, .1) 1 0.1 0.01 0.001 fraction 1e-04 1e-05 1e-06 1e-07 1e-08 0 10 20 30 40 50 60 70 80 90 100 time Figure 3: L(cid:2)L square lattice with L = 101 to 10,001 at s = 0:4. The three left lines correspond to s = 0:1; 0:2; 0:3 from left to right for L = 10;001. The thick horizontal line corresponds to s = 0:5; L = 10;001 and might deviate only for t > 107. 8 2-d lattice. Random updating for L = 20, 30, 50, 70, 100 (sq). Regular updating for L = 30, 100 (*) 100 > <e 10-1 10-2 102 104 time Figure 4: Decay of unstable symmetric solution x = 1=2 for s = 1=2 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 arealsorepresented forregularupdatingforcomparison (dashed lines). Average over 100 samples. 9 1-d lattice. Random updating for L = 100, 200, 400, 1000, 5000 (sq). Regular updating for L = 200, 1000 (*) 100 10-1 <e>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 (*) 100 > <e 10-1 101 102 103 104 time Figure 5: Same as Fig.4but in one (top) and three (bottom) dimensions. For one-dimensional lattice, due tolargertimes fortheseparation fromthe power law decay, and self-averaging for large systems, L = 100;200 are averaged over 1000 samples; L = 400;1000 over 200 samples; and L = 5000 over 50 (random updating). For regular update we average over 100 samples. For three-dimensional lattice we average over 100 samples. 10