Generic Behaviour of Strongly Reinforced Polya Urns : Convergence and Stability

TL;DR

This paper analyzes a class of strongly reinforced Polya urn models inspired by neuronal networks, proving convergence to stable equilibria and confirming conjectures about their stability and structure in a general setting.

Contribution

It establishes the convergence behavior and stability of reinforced urn processes in a broad, general framework, confirming key conjectures from prior work.

Findings

Proves zero probability of convergence to unstable equilibria.

Shows finiteness of equilibrium set under generic conditions.

Confirms full conjecture on stability and convergence in the model.

Abstract

We consider, as proposed and studied in Hofstad et.\ al.\ \cite{HHKR}, a class of graph-based "interacting urn"-type Polya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency' of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved/reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeating this process a very large number of times in a "strong reinforcement regime". Under the most general conditions, we prove in the affirmative a part of the main open conjecture in \cite{HHKR} i.e. the zero probability of convergence of the corresponding "urn process" to any 'unstable' equilibrium. Under very generic conditions, i.e., for an open and dense subset of parameters with full measure, we also prove the full open conjecture by showing the finiteness of the equilibrium set and hence the unit probability of convergence of the "urn process" to some 'stable' equilibrium.