Networks of reinforced stochastic processes: a complete description of the first-order asymptotics
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915333976096768 |
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| author | Aletti, Giacomo Crimaldi, Irene Ghiglietti, Andrea |
| author_facet | Aletti, Giacomo Crimaldi, Irene Ghiglietti, Andrea |
| contents | We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions in order to have some form of almost sure asymptotic synchronization, which could be roughly defined as the almost sure long-run uniformization of the behavior of interacting processes. Specifically, we detect a regime of complete synchronization, where all the processes converge toward the same random variable, a second regime where the system almost surely converges, but there exists no form of almost sure asymptotic synchronization, and another regime where the system does not converge with a strictly positive probability. In this latter case, partitioning the system in cyclic classes according to the period of the interaction matrix, we have an almost sure asymptotic synchronization within the cyclic classes, and, with a strictly positive probability, an asymptotic periodic behavior of these classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07514 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Networks of reinforced stochastic processes: a complete description of the first-order asymptotics Aletti, Giacomo Crimaldi, Irene Ghiglietti, Andrea Probability 60F15, 60K35, 91D30 We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions in order to have some form of almost sure asymptotic synchronization, which could be roughly defined as the almost sure long-run uniformization of the behavior of interacting processes. Specifically, we detect a regime of complete synchronization, where all the processes converge toward the same random variable, a second regime where the system almost surely converges, but there exists no form of almost sure asymptotic synchronization, and another regime where the system does not converge with a strictly positive probability. In this latter case, partitioning the system in cyclic classes according to the period of the interaction matrix, we have an almost sure asymptotic synchronization within the cyclic classes, and, with a strictly positive probability, an asymptotic periodic behavior of these classes. |
| title | Networks of reinforced stochastic processes: a complete description of the first-order asymptotics |
| topic | Probability 60F15, 60K35, 91D30 |
| url | https://arxiv.org/abs/2206.07514 |