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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. 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.
    4. 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.
    5. Irene Crimaldi & Pierre-Yves Louis & Ida Minelli, 2020. "Interacting non-linear reinforced stochastic processes: Synchronization and no-synchronization," Working Papers hal-02910341, HAL.
    6. 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.
    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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