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Genealogies and ages of cultural traits: An application of the theory of duality to the research on cultural evolution

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  • Kobayashi, Yutaka
  • Wakano, Joe Yuichiro
  • Ohtsuki, Hisashi

Abstract

A finite-population, discrete-generation model of cultural evolution is described, in which multiple discrete traits are transmitted independently. In this model, each newborn may inherit a trait from multiple cultural parents. Transmission fails with a positive probability unlike in population genetics. An ancestral process simulating the cultural genealogy of a sample of individuals is derived for this model. This ancestral process, denoted by M−, is shown to be dual to a process M+ describing the change in the frequency of a trait. The age–frequency spectrum is defined as a two-dimensional array whose (i,k) element is the expected number of distinct cultural traits introduced k generations ago and now carried by i individuals in a sample of a particular size n. Numerical calculations reveal that the age–frequency spectrum and related metrics undergo a critical transition from a phase with a moderate number of young, rare traits to a phase with numerous very old, common traits when the expected number of cultural parents per individual exceeds one. It is shown that M+ and M− converge to branching or deterministic processes, depending on the way population size tends to infinity, and these limiting processes bear some duality relationships. The critical behavior of the original processes M+ and M− is explained in terms of a phase transition of the branching processes. Using the results of the limiting processes in combination, we derive analytical formulae that well approximate the age–frequency spectrum and also other metrics.

Suggested Citation

  • Kobayashi, Yutaka & Wakano, Joe Yuichiro & Ohtsuki, Hisashi, 2018. "Genealogies and ages of cultural traits: An application of the theory of duality to the research on cultural evolution," Theoretical Population Biology, Elsevier, vol. 123(C), pages 18-27.
    • Handle: RePEc:eee:thpobi:v:123:y:2018:i:c:p:18-27
    DOI: 10.1016/j.tpb.2018.04.007
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    References listed on IDEAS

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    1. Aoki, Kenichi & Lehmann, Laurent & Feldman, Marcus W., 2011. "Rates of cultural change and patterns of cultural accumulation in stochastic models of social transmission," Theoretical Population Biology, Elsevier, vol. 79(4), pages 192-202.
    2. Strimling, Pontus & Sjöstrand, Jonas & Enquist, Magnus & Eriksson, Kimmo, 2009. "Accumulation of independent cultural traits," Theoretical Population Biology, Elsevier, vol. 76(2), pages 77-83.
    3. Aguilar, Elliot & Ghirlanda, Stefano, 2015. "Modeling the genealogy of a cultural trait," Theoretical Population Biology, Elsevier, vol. 101(C), pages 1-8.
    4. Mano, Shuhei, 2009. "Duality, ancestral and diffusion processes in models with selection," Theoretical Population Biology, Elsevier, vol. 75(2), pages 164-175.
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    Cited by:

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    2. Kobayashi, Yutaka & Kurokawa, Shun & Ishii, Takuya & Wakano, Joe Yuichiro, 2021. "Time to extinction of a cultural trait in an overlapping generation model," Theoretical Population Biology, Elsevier, vol. 137(C), pages 32-45.
    3. Nakamura, Mitsuhiro & Wakano, Joe Yuichiro & Aoki, Kenichi & Kobayashi, Yutaka, 2020. "The popularity spectrum applied to a cross-cultural question," Theoretical Population Biology, Elsevier, vol. 133(C), pages 104-116.

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