Abstract
We study the long-time asymptotics of a network of weakly reinforced Pólya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power , and then this weight is increased by 1.
We show that for on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold .
Acknowledgments
The authors thank both anonymous referees for the careful reading of the manuscript and the constructive feedback. Their comments and suggestions substantially helped to improve the presentation of the material. The authors thank M. Holmes and V. Kleptsyn for illuminating discussions and ideas for future work.
Citation
Yannick Couzinié. Christian Hirsch. "Weakly reinforced Pólya urns on countable networks." Electron. Commun. Probab. 26 1 - 10, 2021. https://doi.org/10.1214/21-ECP404
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