Synchronization in Interacting Reinforced Stochastic Processes

Louis, Pierre‐Yves; Minelli, Ida Germana

doi:10.1007/978-3-319-65558-1_8

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…In [Sah16], this was proven to have consequences for the mean-field interacting system, where different speed of convergence may happen. In relationship with systems of interacting urns, some variations with different kind of urns/reinforcements bias towards one or the other colour, were considered in [LM18].…”

Section: Related Models and Applicative Motivationsmentioning

confidence: 99%

Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

Mirebrahimi

2023

Stochastic Models

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The Pólya urn is the most representative example of a reinforced stochastic process. It leads to a random (non degenerated) time-limit. The Friedman urn is a natural generalization whose almost sure (a.s.) time-limit is not random any more. In this work, in the stream of previous recent works, we introduce a new family of (finite size) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one component-wise, one collective) are tuned through (possibly) different rates behaving asymptotically like n −γ . In the case the reinforcement rates goes like n −1 , these reinforcements are of Pólya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. Different parameter regimes need to be considered.We state two kind of results. First, we study the time-asymptotics and show that L 2 and a.s. convergence always holds. Moreover all the components share the same time-limit (so called synchronization phenomenon). We study the nature of the limit (random/deterministic) according to the parameters' regime considered. Second, we study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into many different rates of convergence. In particular, we identify the regimes where synchronization is faster than convergence towards the shared time-limit.

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Section: Related Models and Applicative Motivationsmentioning

confidence: 99%

Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

Mirebrahimi

2023

Stochastic Models

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“…More generally, cellular automata encode a "unitary time evolution" where no information is lost in a discrete setting with locality properties [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. They are the basis for interesting attempts to obtain quantum mechanics from deterministic physics [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Cellular automaton for spinor gravity in four dimensions

Wetterich

2023

J. Phys.: Conf. Ser.

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Certain fermionic quantum field theories are equivalent to probabilistic cellular automata, with fermionic occupation numbers associated to bits. We construct an automaton that represents a discrete model of spinor gravity in four dimensions. Local Lorentz symmetry is exact on the discrete level and diffeomorphism symmetry emerges in the naive continuum limit. Our setting could serve as a model for quantum gravity if diffeomorphism symmetry is realized in the true continuum limit and suitable collective fields for vierbein and metric acquire non-vanishing expectation values. The discussion of this interesting specific model reveals may key qualitative features of the continuum limit for probabilistic cellular automata. This limit obtains for a very large number of cells if the probabilistic information is sufficiently smooth. It is associated to coarse graining. The automaton property that every bit configuration is updated at every discrete time step to precisely one new bit configuration does no longer hold on the coarse grained level. A coarse grained configuration of occupation numbers can evolve into many different configurations with certain probabilities. This characteristic feature of quantum field theories can come along with the emergence of continuous space-time symmetries.

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“…The analysis of spatial patterns has proven essential to understand ecological system dynamics, and various modelling approaches help ground empirical patterns into ecological theory. Among such approaches, models based on Stochastic Cellular Automata (hereafter SCA), also called Probabilistic or Random Cellular Automata, or Locally-interacting Markov Chains, have been a particularly useful, heuristic and widely used approach (Louis & Nardi 2018; Wolfram 1984). Cellular automata are based on a grid of cells that switch over time between a finite number of states, which capture the entire state space of the system (all possible states assumed as relevant).…”

Section: Introductionmentioning

confidence: 99%

Easy, fast and reproducible Stochastic Cellular Automata with chouca

Génin,

Dupont,

Valencia

et al. 2023

Preprint

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Abstract*Stochastic cellular automata (SCA) are models that describe spatial ecological dynamics using a grid of cells that switch between discrete states over time, depending only on current states (Markov chain processes). They are widely used to understand how small-scale processes scale up to affect ecological dynamics at larger spatial scales, and have been applied to a wide diversity of theoretical and applied problems in all systems, such as arid ecosystems, coral reefs, forests, bacteria, or urban growth.*Despite their wide applications, SCA implementations found in the literature are often ad-hoc, lacking performance and guarantees of correctness. More importantly,de novoimplementation of SCA for each specific system and application represents a major barrier for many practitioners. To provide a unifying, well-tested technical basis to this class of models and facilitate their implementation, we builtchouca. which is an R package that translates intuitive SCA model declarations and expert-based assumptions about the state space system into compiled code and run simulations in a reproducible and efficient way.*choucasupports a wide set of SCA along with deterministic cellular automata, with performance typically two to three orders of magnitude that ofad hocimplementations found in the literature, all while maintaining an intuitive interface in the R environment. Exact and mean-field simulations can be run, and both numerical and graphical results can be easily exported.*Besides providing better reproducibility and accessibility, a fast engine for SCA unlocks novel, computationally intensive statistical approaches, such as simulation-based inference of ecological interactions from field data, which represents by itself an important avenue for research. By providing an easy and efficient entry point to SCAs,choucalowers the bar to the use of this class of models for ecologists, managers and general practitioners, providing a leveled-off reproducible platform while opening novel methodological approaches.

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Synchronization in Interacting Reinforced Stochastic Processes

Cited by 6 publications

References 19 publications

Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

Cellular automaton for spinor gravity in four dimensions

Easy, fast and reproducible Stochastic Cellular Automata with chouca

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