On the Existence and Uniqueness of Fixed Points of Fuzzy Cognitive Maps

Harmati, István; Hatwagner, Miklos F.; Kóczy, László T.

doi:10.1007/978-3-319-91473-2_42

Cited by 21 publications

(7 citation statements)

References 13 publications

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“…In several events involving iterations, such as physics, biology and chemistry phenomena and related knowledge, iteration events are manifestations of repetition in the order or disorder of elements that constitute an iteration [13,14]. Although the cause of iterations as well as their behavior are key topics in the search for iterated events, an important aspect raised in this paper reflects on the discretization of iterated events in terms of knowing how behave diverse iterated functions that generates an event, and this observation serves as a tool to understand the flow with which each function infinitely iterates from a fixed point of dynamic type.…”

Section: Discussionmentioning

confidence: 99%

Measuring nonlinearity by means of static parameters in Bernoulli binary sequences distribution: A brief approach

Telles¹

2020

Int. J. Model. Simul. Sci. Comput.

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This paper analyzes Bernoulli’s binary sequences in the representation of empirical nonlinear events, analyzing the distribution of natural resources, population sizes and other variables that influence the possible outcomes of resource’s usage. Consider the event as a nonlinear system and the metrics of analysis consisting of two dependent random variables 0 and 1, with memory and probabilities in maximum finite or infinite lengths, constant and equal to 1/2 for both variables (stationary process). The expressions of the possible trajectories of metric space represented by each binary parameter remain constant in sequences that are repeated alternating the presence or absence of one of the binary variables at each iteration (symmetric or asymmetric). It was observed that the binary variables [Formula: see text] and [Formula: see text] assume on time [Formula: see text] specific behaviors (geometric variable) that can be used as management tools in discrete and continuous nonlinear systems aiming at the optimization of resource’s usage, nonlinearity analysis and probabilistic distribution of trajectories occurring about random events. In this way, the paper presents a model of detecting fixed-point attractions and its probabilistic distributions at a given population-resource dynamic. This means that coupling oscillations in the event occur when the binary variables [Formula: see text] and [Formula: see text] are limited as a function of time [Formula: see text].

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Section: Discussionmentioning

confidence: 99%

Measuring nonlinearity by means of static parameters in Bernoulli binary sequences distribution: A brief approach

Telles¹

2020

Int. J. Model. Simul. Sci. Comput.

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“…In this section, we use insights reported in [18] to analyze which λ values are not desired in the transfer function, i.e. they lead to a unique fixed-point attractor.…”

Section: A Analyzing Fixed-point Attractorsmentioning

confidence: 99%

“…According to Theorem 3 in [18], if the Frobenius norm of the weight matrix of an FCM is smaller than 4/λ (with λ > 0), then the FCM has one and only one fixed point. Therefore, for any FRCN we have that…”

Section: A Analyzing Fixed-point Attractorsmentioning

confidence: 99%

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Fuzzy-Rough Cognitive Networks: Theoretical Analysis and Simpler Models

Concepción

Nápoles

Grau

et al. 2022

IEEE Trans. Cybern.

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“…Knight et al [8] analyzed FCMs with linear and sigmoid transfer functions. In [9], the authors generalized the findings of [6] to FCMs with arbitrary sigmoid function. All of these studies arrived to the conclusion (although in a different form) that when the parameter of the sigmoid threshold function is small enough, then the FCM converges to a unique fixed point, regardless of the initial activation values.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Fuzzy-Rough Cognitive Networks

Harmati

2021

Symmetry

Self Cite

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Fuzzy-rough cognitive networks (FRCNs) are interpretable recurrent neural networks, primarily designed for solving classification problems. Their structure is simple and transparent, while the performance is comparable to the well-known black-box classifiers. Although there are many applications on fuzzy cognitive maps and recently for FRCNS, only a very limited number of studies discuss the theoretical issues of these models. In this paper, we examine the behaviour of FRCNs viewing them as discrete dynamical systems. It will be shown that their mathematical properties highly depend on the size of the network, i.e., there are structural differences between the long-term behaviour of FRCN models of different size, which may influence the performance of these modelling tools.

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On the Existence and Uniqueness of Fixed Points of Fuzzy Cognitive Maps

Cited by 21 publications

References 13 publications

Measuring nonlinearity by means of static parameters in Bernoulli binary sequences distribution: A brief approach

Measuring nonlinearity by means of static parameters in Bernoulli binary sequences distribution: A brief approach

Fuzzy-Rough Cognitive Networks: Theoretical Analysis and Simpler Models

Dynamics of Fuzzy-Rough Cognitive Networks

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