Fractional Dynamics of Individuals in Complex Networks

Turalska, Malgorzata; West, Bruce J.

doi:10.3389/fphy.2018.00110

Cited by 15 publications

(25 citation statements)

References 26 publications

(26 reference statements)

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“…However, we use the term "quantum-like coherence" rather than quantum coherence. We note also that Turalska and West [25] used a model of interacting units to mimic a decision making system and found that the behavior of the single units is very well described by Equation (8). This implies that the Caputo fractional derivative may have an important role to play in understanding the conjunction fallacy, that being, given two events A and B the probability P(A) is always greater than the conjoined probability P(A ∪ B).…”

Section: Fractional Diffusion Equationmentioning

confidence: 84%

“…Further research must be done to confirm these results and to strengthen the foundations for the rigorous theory, which may overlap with research on chaos-assisted tunneling. The Caputo fractional derivatives are manifestations of crucial events, a recent example of the relevance of which to complexity in network dynamics is found in [ 25 ]. In it, Turalska and West studied a social network of interacting individuals at criticality and found that the global complex network generates crucial events.…”

Section: Resultsmentioning

confidence: 99%

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Caputo Fractional Derivative and Quantum-Like Coherence

Culbreth

Bologna

West

et al. 2021

Entropy

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We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

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Section: Fractional Diffusion Equationmentioning

confidence: 84%

Section: Resultsmentioning

confidence: 99%

Caputo Fractional Derivative and Quantum-Like Coherence

Culbreth

Bologna

West

et al. 2021

Entropy

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“…Here, we follow the discussion of Turalska and West [ 21 ]. The idea of different clocks telling different times arises naturally in physics; the linear transformation of Lorentz in relativistic physics being a familiar example.…”

Section: Introductionmentioning

confidence: 99%

“…If each tick of the discrete clock n is considered to be an event, the relation between operational and chronological time is given by the waiting time PDF of those events in chronological time

. Assuming a renewal property for events, as given by a chain condition (convolution) from renewal theory in Section 2.1 , one can relate operational to chronological time [ 21 ]:

Every tick of the operational clock is an event, which in the chronological time occurs at time intervals drawn from the renewal waitin -time PDF. This randomness entails the sum over all events and the result is an average over many realizations of the transformation.…”

Section: Introductionmentioning

confidence: 99%

Sir Isaac Newton Stranger in a Strange Land

West

2020

Entropy

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The theme of this essay is that the time of dominance of Newton’s world view in science is drawing to a close. The harbinger of its demise was the work of Poincaré on the three-body problem and its culmination into what is now called chaos theory. The signature of chaos is the sensitive dependence on initial conditions resulting in the unpredictability of single particle trajectories. Classical determinism has become increasingly rare with the advent of chaos, being replaced by erratic stochastic processes. However, even the probability calculus could not withstand the non-Newtonian assault from the social and life sciences. The ordinary partial differential equations that traditionally determined the evolution of probability density functions (PDFs) in phase space are replaced with their fractional counterparts. Allometry relation is proven to result from a system’s complexity using exact solutions for the PDF of the Fractional Kinetic Theory (FKT). Complexity theory is shown to be incompatible with Newton’s unquestioning reliance on an absolute space and time upon which he built his discrete calculus.

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“…The fractional calculus (FC) is a branch of mathematical analysis, which deals with the non-integer integral and derivative operators. The application of the FC has been extensively expanded in different fields of the basic and engineering sciences [2][3][4][5][6][7][8][9]. Over the past few decades, the classical mechanics has been extended by using the new aspects of the FC.…”

Section: Introductionmentioning

confidence: 99%

A New Feature of the Fractional Euler–Lagrange Equations for a Coupled Oscillator Using a Nonsingular Operator Approach

et al. 2019

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In this new work, the free motion of a coupled oscillator is investigated. First, a fully description of the system under study is formulated by considering its classical Lagrangian, and as a result, the classical Euler-Lagrange equations of motion are constructed. After this point, we extend the classical Lagrangian in fractional sense, and thus, the fractional Euler-Lagrange equations of motion are derived. In this new formulation, we consider a recently introduced fractional operator with Mittag-Leffler non-singular kernel. We also present an efficient numerical method for solving the latter equations in a proper manner. Due to this new powerful technique, we are able to obtain remarkable physical thinks; indeed, we indicate that the complex behavior of many physical systems is realistically demonstrated via the fractional calculus modeling. Finally, we report our numerical findings to verify the theoretical analysis.

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Fractional Dynamics of Individuals in Complex Networks

Cited by 15 publications

References 26 publications

Caputo Fractional Derivative and Quantum-Like Coherence

Caputo Fractional Derivative and Quantum-Like Coherence

Sir Isaac Newton Stranger in a Strange Land

A New Feature of the Fractional Euler–Lagrange Equations for a Coupled Oscillator Using a Nonsingular Operator Approach

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