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Title: unleashing the potential of fractional calculus in graph neural networks with frond.
Abstract: We introduce the FRactional-Order graph Neural Dynamical network (FROND), a new continuous graph neural network (GNN) framework. Unlike traditional continuous GNNs that rely on integer-order differential equations, FROND employs the Caputo fractional derivative to leverage the non-local properties of fractional calculus. This approach enables the capture of long-term dependencies in feature updates, moving beyond the Markovian update mechanisms in conventional integer-order models and offering enhanced capabilities in graph representation learning. We offer an interpretation of the node feature updating process in FROND from a non-Markovian random walk perspective when the feature updating is particularly governed by a diffusion process. We demonstrate analytically that oversmoothing can be mitigated in this setting. Experimentally, we validate the FROND framework by comparing the fractional adaptations of various established integer-order continuous GNNs, demonstrating their consistently improved performance and underscoring the framework's potential as an effective extension to enhance traditional continuous GNNs. The code is available at \url{ this https URL }.
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Fractional calculus for power functions and eigenvalues of the fractional Laplacian
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- Published: 29 September 2012
- Volume 15 , pages 536–555, ( 2012 )
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- Bartłlomiej Dyda 1 , 2
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We calculate the fractional Laplacian Δ α /2 for functions of the form u ( x ) = (1 − | x | 2 ) p + and v ( x ) = x d u ( x ). As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.
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Geometric properties of certain analytic functions associated with generalized fractional integral operators
More properties of the proportional fractional integrals and derivatives of a function with respect to another function, fractional laplace operator and meijer g-function.
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Faculty of Mathematics, University of Bielefeld, Postfach 10 01 31, D-33501, Bielefeld, Germany
Bartłlomiej Dyda
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland
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Dyda, B. Fractional calculus for power functions and eigenvalues of the fractional Laplacian. fcaa 15 , 536–555 (2012). https://doi.org/10.2478/s13540-012-0038-8
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Received : 27 March 2012
Published : 29 September 2012
Issue Date : December 2012
DOI : https://doi.org/10.2478/s13540-012-0038-8
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RESEARCH PAPER FRACTIONAL CALCULUS FOR POWER FUNCTIONS AND EIGENVALUES OF THE FRACTIONAL LAPLACIAN Bartlomiej Dyda Abstract We calculate the fractional Laplacian Δα/2 for functions of the form u(x)=(1−|x|2)p + and v(x)=x du(x). As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.