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Interacting urns on a finite directed graph

Part of: Special processes Operations research and management science Limit theorems

Published online by Cambridge University Press:  23 September 2022

Gursharn Kaur  and
Neeraja Sahasrabudhe
Gursharn Kaur*
Affiliation:
University of Virginia
Neeraja Sahasrabudhe*
Affiliation:
Indian Institute of Science Education and Research
*
*Postal address: Biocomplexity Institute and Initiative, 994 Research Park Boulevard, University of Virginia, Charlottesville, VA 22911. Email address: gursharn.kaur24@gmail.com
**Postal address: Department of Mathematical Sciences, IISER Mohali, Knowledge City, Sector 81, SAS Nagar, Manauli PO 140306. Email address: neeraja@iisermohali.ac.in
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Abstract

We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

Keywords

ReinforcementPólya urnsFriedman urnslimit theoremsopinion dynamics on networkssynchronisation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F05: Central limit and other weak theorems 90B15: Network models, stochastic
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 166 - 188
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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