Strongly reinforced Pólya urns with graph-based competition
| Autor: | Wioletta M. Ruszel, Mark Holmes, Remco van der Hofstad, Alexey Kuznetsov |
|---|---|
| Přispěvatelé: | Stochastic Operations Research, Probability |
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
Class (set theory) 37C10 Stability (learning theory) Stable equilibria Context (language use) Astrophysics::Cosmology and Extragalactic Astrophysics 0102 computer and information sciences Pólya um 01 natural sciences Pólya urn Competition (economics) Combinatorics 010104 statistics & probability Reinforcement model 0101 mathematics stable equilibria Astrophysics::Galaxy Astrophysics Mathematics stochastic approximation algorithm Toy model Graph based Random walk 010201 computation theory & mathematics 60K35 Graph (abstract data type) Statistics Probability and Uncertainty |
| Zdroj: | Ann. Appl. Probab. 26, no. 4 (2016), 2494-2539 The Annals of Applied Probability, 26(4), 2494-2539. Institute of Mathematical Statistics Annals of Applied Probability, 26(4), 2494. Institute of Mathematical Statistics |
| ISSN: | 1050-5164 |
| Popis: | We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_{t}$ of colours (independently of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $\alpha>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph; a context which is a toy model for the formation and reinforcement of neural connections. We conjecture that for any graph $G$ and all $\alpha$ sufficiently large, the set of stable equilibria is supported on so-called whisker-forests, which are forests whose components have diameter between 1 and 3. |
| Databáze: | OpenAIRE |
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