Interacting Urns on Directed Networks with Node-Dependent Sampling and Reinforcement

• Main Authors: Kaur, Gursharn, Sahasrabudhe, Neeraja
• Format: Journal Article
• Language: English
• Published: 04-12-2023
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2312.02096

We consider interacting urns on a finite directed network, where both
sampling and reinforcement processes depend on the nodes of the network. This
extends previous research by incorporating node-dependent sampling
(preferential or de-preferential) and reinforcement. We classify the
reinforcement schemes and the networks on which the proportion of balls of
either colour in each urn converges almost surely to a deterministic limit. We
show that in case the reinforcement at all nodes is of P\'olya-type, the
limiting behaviour is very different from the node-independent sampling and a
deterministic limit exists for certain networks classified by the distribution
of preferential and de-preferential nodes across the network. We also
investigate conditions for achieving synchronisation of the colour proportions
across the urns. Further, we analyse fluctuations around the limit, under
specific conditions on the reinforcement matrices and network structure.