Interacting Urn Schemes on Infinite Graphs: A Representation Approach
Abstract
Interacting urn schemes have emerged as an important framework for studying reinforcement phenomena in interacting stochastic systems (see, e.g., \cite{launay2011interacting,launay2012generalized,WARM2014,aletti2017interacting,Synchronisation-2,kaur2023interacting,kaur2023interacting2}). Most existing results, however, are restricted to finite systems. In this article, we investigate interacting urn schemes involving infinitely many urns, where the interactions are naturally described by a directed graph. The reinforcement or drawing mechanism at each urn depends on the compositions of its in-neighboring urns. To analyze the asymptotic behavior of these systems, we develop a representation of the urn dynamics in terms of branching Markov processes evolving on a class of random forests introduced in this article. This extends the representation approach of \cite{bandyopadhyay2022new}, originally developed for single Pólya urn schemes, to interacting urn schemes on locally finite directed graphs. Using this representation, we establish almost sure convergence of the color proportions under standard conditions on the reinforcement matrix or kernel. In particular, we show that the convergence holds independently of the topology of the underlying graph, and hence our results also apply to undirected graphs, where the interactions are bidirectional. We further investigate measure-valued generalizations and other related reinforced stochastic processes, illustrating the broad applicability of the representation approach. Finally, we discuss connections with self-organized criticality.
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BibTeX
@misc{bandyopadhyay2026interacting,
title = {{Interacting Urn Schemes on Infinite Graphs: A Representation Approach}},
author = {Antar Bandyopadhyay and Deborshi Das},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08611},
eprint = {2611.08611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08611},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Interacting Urn Schemes on Infinite Graphs: A Representation Approach
AU - Antar Bandyopadhyay
AU - Deborshi Das
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08611
UR - https://arxiv.org/abs/2611.08611
AB - Interacting urn schemes have emerged as an important framework for studying reinforcement phenomena in interacting stochastic systems (see, e.g., \cite{launay2011interacting,launay2012generalized,WARM2014,aletti2017interacting,Synchronisation-2,kaur2023interacting,kaur2023interacting2}). Most existing results, however, are restricted to finite systems. In this article, we investigate interacting urn schemes involving infinitely many urns, where the interactions are naturally described by a directed graph. The reinforcement or drawing mechanism at each urn depends on the compositions of its in-neighboring urns. To analyze the asymptotic behavior of these systems, we develop a representation of the urn dynamics in terms of branching Markov processes evolving on a class of random forests introduced in this article. This extends the representation approach of \cite{bandyopadhyay2022new}, originally developed for single Pólya urn schemes, to interacting urn schemes on locally finite directed graphs. Using this representation, we establish almost sure convergence of the color proportions under standard conditions on the reinforcement matrix or kernel. In particular, we show that the convergence holds independently of the topology of the underlying graph, and hence our results also apply to undirected graphs, where the interactions are bidirectional. We further investigate measure-valued generalizations and other related reinforced stochastic processes, illustrating the broad applicability of the representation approach. Finally, we discuss connections with self-organized criticality.
ER -