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Asymptotics of a matrix valued Markov chain arising in sociology

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  • Bonacich, Phillip
  • Liggett, Thomas M.

Abstract

We consider a discrete time Markov chain whose state space is the set of all NxN stochastic matrices with zero diagonal entries. This chain models the evolution of relationships among N individuals who exchange gifts according to probabilities determined by previous exchanges. We determine the stable equilibria for this chain, and prove convergence to a mixture of these. In particular, we show that for generic initial states, the chain converges to a randomly chosen set of constellations made up of disjoint stars. Each star has a center, which is the recipient of all gifts from the other individuals in that star, while the center distributes his gifts only to members of his own star.

Suggested Citation

  • Bonacich, Phillip & Liggett, Thomas M., 2003. "Asymptotics of a matrix valued Markov chain arising in sociology," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 155-171, March.
  • Handle: RePEc:eee:spapps:v:104:y:2003:i:1:p:155-171
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    Citations

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    Cited by:

    1. Liggett, Thomas M. & Rolles, Silke W. W., 2004. "An infinite stochastic model of social network formation," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 65-80, September.
    2. Brian Skyrms & Robin Pemantle, 2004. "Learning to Network," Levine's Bibliography 122247000000000436, UCLA Department of Economics.
    3. Argiento, Raffaele & Pemantle, Robin & Skyrms, Brian & Volkov, Stanislav, 2009. "Learning to signal: Analysis of a micro-level reinforcement model," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 373-390, February.
    4. Irene Crimaldi & Pierre-Yves Louis & Ida Minelli, 2020. "Interacting non-linear reinforced stochastic processes: Synchronization and no-synchronization," Working Papers hal-02910341, HAL.
    5. Georgios Chasparis & Jeff Shamma & Anders Rantzer, 2015. "Nonconvergence to saddle boundary points under perturbed reinforcement learning," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(3), pages 667-699, August.
    6. Matthias Greiff, 2013. "Rewards and the private provision of public goods on dynamic networks," Journal of Evolutionary Economics, Springer, vol. 23(5), pages 1001-1021, November.
    7. Pemantle, Robin & Skyrms, Brian, 2004. "Network formation by reinforcement learning: the long and medium run," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 315-327, November.

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