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Urns with Multiple Drawings and Graph-Based Interaction

Author

Listed:
  • Yogesh Dahiya

    (Indian Institute of Science Education and Research)

  • Neeraja Sahasrabudhe

    (Indian Institute of Science Education and Research)

Abstract

Consider a finite undirected graph and place an urn with balls of two colours at each vertex. At every discrete time step, for each urn, a fixed number of balls are drawn from that same urn with probability p and from a randomly chosen neighbour of that urn with probability $$1-p$$ 1 - p . Based on what is drawn, the urns then reinforce themselves or their neighbours. For every ball of a given colour in the sample, in case of Pólya-type reinforcement, a constant multiple of balls of that colour is added while in case of Friedman-type reinforcement, balls of the other colour are reinforced. These different choices for reinforcement give rise to multiple models. In this paper, we study the convergence of the fraction of balls of either colour across urns for all of these models. We show that in most cases the urns synchronize, that is, the fraction of balls of either colour in each urn converges to the same limit almost surely. A different kind of asymptotic behaviour is observed on bipartite graphs. We also prove similar results for the case of finite directed graphs.

Suggested Citation

  • Yogesh Dahiya & Neeraja Sahasrabudhe, 2024. "Urns with Multiple Drawings and Graph-Based Interaction," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3283-3316, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01365-x
    DOI: 10.1007/s10959-024-01365-x
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    References listed on IDEAS

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    1. Crimaldi, Irene & Dai Pra, Paolo & Louis, Pierre-Yves & Minelli, Ida G., 2019. "Synchronization and functional central limit theorems for interacting reinforced random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 70-101.
    2. Tadić, Vladislav B., 2015. "Convergence and convergence rate of stochastic gradient search in the case of multiple and non-isolated extrema," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1715-1755.
    3. Aguech Rafik & Lasmar Nabil & Selmi Olfa, 2019. "A generalized urn with multiple drawing and random addition," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 389-408, April.
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