Computability and Dynamical Systems

Buescu, Jorge; Graça, Daniel S.; Zhong, Ning

doi:10.1007/978-3-642-11456-4_11

Cited by 3 publications

(1 citation statement)

References 61 publications

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“…A final outstanding issue is determining when the occupation vector of the spacey random walk will converge. We suspect that this problem is undecidable as similar results have been shown for general dynamical systems [Buescu et al, 2011]. One approach for this problem is to show that the spacey random walker can simulate a Turing machine at which point determining convergence is equivalent to the halting problem (see the work of Moore [1990] for a similar approach).…”

Section: Zeroth-ordermentioning

confidence: 87%

The Spacey Random Walk: A Stochastic Process for Higher-Order Data

Benson¹,

Gleich²,

Lim³

2017

SIAM Rev.

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Random walks are a fundamental model in applied mathematics and are a common example of a Markov chain. The limiting stationary distribution of the Markov chain represents the fraction of the time spent in each state during the stochastic process. A standard way to compute this distribution for a random walk on a finite set of states is to compute the Perron vector of the associated transition matrix. There are algebraic analogues of this Perron vector in terms of transition probability tensors of higher-order Markov chains. These vectors are nonnegative, have dimension equal to the dimension of the state space, and sum to one and are derived by making an algebraic substitution in the equation for the joint-stationary distribution of a higher-order Markov chains. Here, we present the spacey random walk, a non-Markovian stochastic process whose stationary distribution is given by the tensor eigenvector. The process itself is a vertex-reinforced random walk, and its discrete dynamics are related to a continuous dynamical system. We analyze the convergence properties of these dynamics and discuss numerical methods for computing the stationary distribution. Finally, we provide several applications of the spacey random walk model in population genetics, ranking, and clustering data, and we use the process to analyze taxi trajectory data in New York. This example shows definite non-Markovian structure.

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Section: Zeroth-ordermentioning

confidence: 87%

The Spacey Random Walk: A Stochastic Process for Higher-Order Data

Benson¹,

Gleich²,

Lim³

2017

SIAM Rev.

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Elastic Multi-scale Mechanisms: Computation and Biological Evolution

Ochoa¹

2017

J Mol Evol

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Explanations based on low-level interacting elements are valuable and powerful since they contribute to identify the key mechanisms of biological functions. However, many dynamic systems based on low-level interacting elements with unambiguous, finite, and complete information of initial states generate future states that cannot be predicted, implying an increase of complexity and open-ended evolution. Such systems are like Turing machines, that overlap with dynamical systems that cannot halt. We argue that organisms find halting conditions by distorting these mechanisms, creating conditions for a constant creativity that drives evolution. We introduce a modulus of elasticity to measure the changes in these mechanisms in response to changes in the computed environment. We test this concept in a population of predators and predated cells with chemotactic mechanisms and demonstrate how the selection of a given mechanism depends on the entire population. We finally explore this concept in different frameworks and postulate that the identification of predictive mechanisms is only successful with small elasticity modulus.

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Towards a Constructive Framework for Control Theory

Osinenko

2022

IEEE Control Syst. Lett.

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Computability and Dynamical Systems

Cited by 3 publications

References 61 publications

The Spacey Random Walk: A Stochastic Process for Higher-Order Data

The Spacey Random Walk: A Stochastic Process for Higher-Order Data

Elastic Multi-scale Mechanisms: Computation and Biological Evolution

Towards a Constructive Framework for Control Theory

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