Nonlinear voter models: the transition from invasion to coexistence

F Schweitzer, Laxmidhar Behera

    Research output: Contribution to journalArticle

    42 Citations (Scopus)

    Abstract

    In nonlinear voter models the transitions between two states depend in a nonlinear manneron the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions fromthe mean-field phase diagram and are confirmed by extensive computer simulations of the microscopicdynamics.PACS. 87.23.Cc
    LanguageEnglish
    Pages301-318
    JournalEuropean Physical Journal B: Condensed Matter and Complex Systems
    Volume67
    Issue number3
    DOIs
    Publication statusPublished - 2009

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    Phase diagrams
    phase diagrams
    Picture archiving and communication systems
    approximation
    computerized simulation
    nonlinearity
    Computer simulation
    predictions
    Direction compound

    Keywords

    • Population dynamics and ecological pattern formation
    • Dynamics of social
    • systems

    Cite this

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    title = "Nonlinear voter models: the transition from invasion to coexistence",
    abstract = "In nonlinear voter models the transitions between two states depend in a nonlinear manneron the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions fromthe mean-field phase diagram and are confirmed by extensive computer simulations of the microscopicdynamics.PACS. 87.23.Cc",
    keywords = "Population dynamics and ecological pattern formation, Dynamics of social, systems",
    author = "F Schweitzer and Laxmidhar Behera",
    note = "Reference text: 1. D. Abrams, S. Strogatz, Nature 424, 900 (2003) 2. P.S. Albin, The Analysis of Complex Socioeconomic Systems (Lexington Books, London, 1975) 3. J. Antonovics, P. Kareiva, Philos. Trans. R. Soc. London B 319, 601 (1988) 4. L. Behera, F. Schweitzer, Int. J. Mod. Phys. C 14, 1331 (2003) 5. E. Ben-Naim, L. Frachebourg, P.L. Krapivsky, Phys. Rev. E 53, 3078 (1996) 6. E. Ben-Naim, P. Krapivsky, S. Redner, Physica D: Nonlinear Phenomena 183, 190 (2003) 7. A.T. Bernardes, D. Stauffer, J. Kertesz, Eur. Phys. J. B 25, 123 (2002) 8. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. (2008) http://arxiv.org/abs/0710.3256 9. C. Castellano, V. Loreto, A. Barrat, F. Cecconi, D. Parisi, Phys. Rev. E 71, 066107 (2005) 10. C. Castellano, D. Vilone, A. Vespignani, Europhys. Lett. 63, 153 (2003) 11. X. Castello, V. Egu´ıluz, M. San Miguel, New J. Phys. 8, 308 (2006) 12. R.N. Costa Filho, M.P. Almeida, J.S. Andrade, J.E. Moreira, Phys. Rev. E 60, 1067 (1999) 13. J.T. Cox, D. Griffeath, Annals of Probability 14, 347 (1986) 14. L. Dall’Asta, C. Castellano, Europhys. Lett. 77, 60005 (2007) 15. M. De Oliveira, J. Mendes,M. Santos, J. Phys. A 26, 2317 (1993) 16. I. Dornic, H. Chat´e, J. Chave, H. Hinrichsen, Phys. Rev. Lett. 87, 045701 (2001) 17. J. Drouffe, C. Godr`eche, J. Phys. A: Mathematical and General 32, 249 (1999) 18. R. Durrett, S. Levin, Theoretical Population Biology 46, 363 (1994) 19. L. Frachebourg, P. Krapivsky, Phys. Rev. E 53, R3009 (1996) 20. R.A. Holley, T.M. Liggett, Annals of Probability 3, 643 (1975) 21. J. Ho�lyst, K. Kacperski, F. Schweitzer, Physica A 285, 199 (2000) 22. T.H. Keitt, M.A. Lewis, R.D. Holt, The American Naturalist 157, 203 (2001) 23. B.E. Kendall, O.N. Bj{\o}rnstad, J. Bascompte, T.H. Keitt, W.F. Fagan, The American Naturalist 155, 628 (2000) 24. M. Kimura, G.H. Weiss, Genetics 49, 313 (1964) 25. P. Krapivsky, Phys. Rev. A 45, 1067 (1992) 26. P.L. Krapivsky, S. Redner, Phys. Rev. Lett. 90, 238701 (2003) 27. T.M. Liggett, Annals of Probability 22, 764 (1994) 28. T.M. Liggett, Stochastic Interacting Systems, Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 1999), Vol. 342 29. J. Molofsky, R. Durrett, J. Dushoff, D. Griffeath, S. Levin, Theoretical Population Biology 55, 270 (1999) 30. C. Moore, J. Stat. Phys. 88, 795 (1997) 31. H. M¨uhlenbein, R. H¨ons, Advances in Complex Systems 5, 301 (2002) 32. M. Nakamaru, H. Matsuda, Y. Iwasa, J. Theor. Biology 184, 65 (1997) 33. C. Neuhauser, Theoretical Population Biology 56, 203 (1999) 34. A. Nowak, M. Kus, J. Urbaniak, T. Zarycki, Physica A 287, 613 (2000) 35. N.A. Oomes, D. Griffeath, C. Moore, New Constructions in cellular automata (Oxford Universitiy Press, 2002), pp. 207–230 36. S.W. Pacala, J.A. Silander, Jr., The American Naturalist 125, 385 (1985) 37. S. Redner, A guide to first-passage processes (Cambridge University Press, Cambridge, 2001) 38. F.J. Rohlf, G.D. Schnell, The American Naturalist 105, 295 (1971) 39. T. Schelling, Am. Econ. Rev. 59, 488 (1969) 40. F. Schweitzer, Brownian Agents and Active Particles. Collective Dynamics in the Natural and Social Sciences, Springer Series in Synergetics (Springer, Berlin, 2003) 41. F. Schweitzer, L. Behera, H. M¨uhlenbein, Advances in Complex Systems 5, 269 (2002) 42. F. Schweitzer, J. Ho�lyst, Eur. Phys. J. B 15, 723 (2000) 43. F. Slanina, H. Lavicka, Eur. Phys. J. B 35, 279 (2003) 44. V. Sood, S. Redner, Phys. Rev. Lett. 94, 178701 (2005) 45. H.-U. Stark, C.J. Tessone, F. Schweitzer, Phys. Rev. Lett. 101, 018701 (2008) 46. H.-U. Stark, C.J. Tessone, F. Schweitzer, Advances in Complex Systems 11, 87 (2008) 47. K. Suchecki, V.M. Egu´ıluz, M. San Miguel, Europhys. Lett. 69, 228 (2005) 48. K. Suchecki, V.M. Egu´ıluz, M. San Miguel, Phys. Rev. E 72, 036132 (2005) 49. G. Szab´o, T. Antal, P. Szab´o, M. Droz, Phys. Rev. E 62, 1095 (2000) 50. F. Vazquez, V.M. Eguiluz, M.S. Miguel, Phys. Rev. Lett. 100, 108702 (2008) 51. F. Vazquez, C. Lopez, Phys. Rev. E 78, 061127 (2008) 52. W. Weidlich, J. Mathematical Sociology 18, 267 (1994",
    year = "2009",
    doi = "10.1140/epjb/e2009-00001-3",
    language = "English",
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    pages = "301--318",
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    }

    Nonlinear voter models: the transition from invasion to coexistence. / Schweitzer, F; Behera, Laxmidhar.

    In: European Physical Journal B: Condensed Matter and Complex Systems, Vol. 67, No. 3, 2009, p. 301-318.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Nonlinear voter models: the transition from invasion to coexistence

    AU - Schweitzer, F

    AU - Behera, Laxmidhar

    N1 - Reference text: 1. D. Abrams, S. Strogatz, Nature 424, 900 (2003) 2. P.S. Albin, The Analysis of Complex Socioeconomic Systems (Lexington Books, London, 1975) 3. J. Antonovics, P. Kareiva, Philos. Trans. R. Soc. London B 319, 601 (1988) 4. L. Behera, F. Schweitzer, Int. J. Mod. Phys. C 14, 1331 (2003) 5. E. Ben-Naim, L. Frachebourg, P.L. Krapivsky, Phys. Rev. E 53, 3078 (1996) 6. E. Ben-Naim, P. Krapivsky, S. Redner, Physica D: Nonlinear Phenomena 183, 190 (2003) 7. A.T. Bernardes, D. Stauffer, J. Kertesz, Eur. Phys. J. B 25, 123 (2002) 8. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. (2008) http://arxiv.org/abs/0710.3256 9. C. Castellano, V. Loreto, A. Barrat, F. Cecconi, D. Parisi, Phys. Rev. E 71, 066107 (2005) 10. C. Castellano, D. Vilone, A. Vespignani, Europhys. Lett. 63, 153 (2003) 11. X. Castello, V. Egu´ıluz, M. San Miguel, New J. Phys. 8, 308 (2006) 12. R.N. Costa Filho, M.P. Almeida, J.S. Andrade, J.E. Moreira, Phys. Rev. E 60, 1067 (1999) 13. J.T. Cox, D. Griffeath, Annals of Probability 14, 347 (1986) 14. L. Dall’Asta, C. Castellano, Europhys. Lett. 77, 60005 (2007) 15. M. De Oliveira, J. Mendes,M. Santos, J. Phys. A 26, 2317 (1993) 16. I. Dornic, H. Chat´e, J. Chave, H. Hinrichsen, Phys. Rev. Lett. 87, 045701 (2001) 17. J. Drouffe, C. Godr`eche, J. Phys. A: Mathematical and General 32, 249 (1999) 18. R. Durrett, S. Levin, Theoretical Population Biology 46, 363 (1994) 19. L. Frachebourg, P. Krapivsky, Phys. Rev. E 53, R3009 (1996) 20. R.A. Holley, T.M. Liggett, Annals of Probability 3, 643 (1975) 21. J. Ho�lyst, K. Kacperski, F. Schweitzer, Physica A 285, 199 (2000) 22. T.H. Keitt, M.A. Lewis, R.D. Holt, The American Naturalist 157, 203 (2001) 23. B.E. Kendall, O.N. Bjørnstad, J. Bascompte, T.H. Keitt, W.F. Fagan, The American Naturalist 155, 628 (2000) 24. M. Kimura, G.H. Weiss, Genetics 49, 313 (1964) 25. P. Krapivsky, Phys. Rev. A 45, 1067 (1992) 26. P.L. Krapivsky, S. Redner, Phys. Rev. Lett. 90, 238701 (2003) 27. T.M. Liggett, Annals of Probability 22, 764 (1994) 28. T.M. Liggett, Stochastic Interacting Systems, Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 1999), Vol. 342 29. J. Molofsky, R. Durrett, J. Dushoff, D. Griffeath, S. Levin, Theoretical Population Biology 55, 270 (1999) 30. C. Moore, J. Stat. Phys. 88, 795 (1997) 31. H. M¨uhlenbein, R. H¨ons, Advances in Complex Systems 5, 301 (2002) 32. M. Nakamaru, H. Matsuda, Y. Iwasa, J. Theor. Biology 184, 65 (1997) 33. C. Neuhauser, Theoretical Population Biology 56, 203 (1999) 34. A. Nowak, M. Kus, J. Urbaniak, T. Zarycki, Physica A 287, 613 (2000) 35. N.A. Oomes, D. Griffeath, C. Moore, New Constructions in cellular automata (Oxford Universitiy Press, 2002), pp. 207–230 36. S.W. Pacala, J.A. Silander, Jr., The American Naturalist 125, 385 (1985) 37. S. Redner, A guide to first-passage processes (Cambridge University Press, Cambridge, 2001) 38. F.J. Rohlf, G.D. Schnell, The American Naturalist 105, 295 (1971) 39. T. Schelling, Am. Econ. Rev. 59, 488 (1969) 40. F. Schweitzer, Brownian Agents and Active Particles. Collective Dynamics in the Natural and Social Sciences, Springer Series in Synergetics (Springer, Berlin, 2003) 41. F. Schweitzer, L. Behera, H. M¨uhlenbein, Advances in Complex Systems 5, 269 (2002) 42. F. Schweitzer, J. Ho�lyst, Eur. Phys. J. B 15, 723 (2000) 43. F. Slanina, H. Lavicka, Eur. Phys. J. B 35, 279 (2003) 44. V. Sood, S. Redner, Phys. Rev. Lett. 94, 178701 (2005) 45. H.-U. Stark, C.J. Tessone, F. Schweitzer, Phys. Rev. Lett. 101, 018701 (2008) 46. H.-U. Stark, C.J. Tessone, F. Schweitzer, Advances in Complex Systems 11, 87 (2008) 47. K. Suchecki, V.M. Egu´ıluz, M. San Miguel, Europhys. Lett. 69, 228 (2005) 48. K. Suchecki, V.M. Egu´ıluz, M. San Miguel, Phys. Rev. E 72, 036132 (2005) 49. G. Szab´o, T. Antal, P. Szab´o, M. Droz, Phys. Rev. E 62, 1095 (2000) 50. F. Vazquez, V.M. Eguiluz, M.S. Miguel, Phys. Rev. Lett. 100, 108702 (2008) 51. F. Vazquez, C. Lopez, Phys. Rev. E 78, 061127 (2008) 52. W. Weidlich, J. Mathematical Sociology 18, 267 (1994

    PY - 2009

    Y1 - 2009

    N2 - In nonlinear voter models the transitions between two states depend in a nonlinear manneron the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions fromthe mean-field phase diagram and are confirmed by extensive computer simulations of the microscopicdynamics.PACS. 87.23.Cc

    AB - In nonlinear voter models the transitions between two states depend in a nonlinear manneron the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions fromthe mean-field phase diagram and are confirmed by extensive computer simulations of the microscopicdynamics.PACS. 87.23.Cc

    KW - Population dynamics and ecological pattern formation

    KW - Dynamics of social

    KW - systems

    U2 - 10.1140/epjb/e2009-00001-3

    DO - 10.1140/epjb/e2009-00001-3

    M3 - Article

    VL - 67

    SP - 301

    EP - 318

    JO - European Physical Journal B: Condensed Matter and Complex Systems

    T2 - European Physical Journal B: Condensed Matter and Complex Systems

    JF - European Physical Journal B: Condensed Matter and Complex Systems

    SN - 1434-6028

    IS - 3

    ER -