On Frink's Type Metrization of Weighted Graphs

Main Article Content

Mara Florencia Acosta
Hugo Aimar
Ivana Gomez

Abstract

Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric d(x, y) between the vertices x and y of an anity weighted undirected graph.

Keywords:
Metrization, uniform spaces, weighted graphs.

Article Details

How to Cite
Acosta, M. F., Aimar, H., & Gomez, I. (2021). On Frink’s Type Metrization of Weighted Graphs. Asian Research Journal of Mathematics, 17(1), 26-37. https://doi.org/10.9734/arjom/2021/v17i130262
Section
Original Research Article

References

Mark Tygert, Joan Bruna, Soumith Chintala, Yann LeCun, Serkan Piantino, Arthur Szlam.

A mathematical motivation for complex-valued convolutional networks. Neural Computation.;28(5):815-825. PMID: 26890348.

Yann LeCun, Yoshua Bengio, Georey E. Hinton. Deep learning. Nat. 2015;521(7553):436-444.

Karen Simonyan, Andrew Zisserman. Very deep convolutional networks for large-scale image

recognition. In Yoshua Bengio, Yann LeCun, Editors. 3rd International Conference on Learning

Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings; 2015.

Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner F, Zucker SW. Geometric diusions as a tool for harmonic analysis and structure denition of data: Diusion maps.

Proceedings of the National Academy of Sciences. 2005;102(21):7426-7431.

Ronald R. Coifman and Stephane Lafon. Diusion maps. Appl. Comput. Harmon. Anal. ;21:5-30.

Frink AH. Distance functions and the metrization problem. Bull. Amer. Math. Soc. ;43(2):133-142.

Chittenden EW. On the metrization problem and related problems in the theory of abstract sets. Bull. Amer. Math. Soc. 1927;33:13-34.

John L. Kelley. General topology. Springer-Verlag. New York-Berlin; 1975. Reprint of the 1955

edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27.

Roberto A. Macas, Carlos Segovia. Lipschitz functions on spaces of homogeneous type. Adv. in Math. 1979;33(3):257-270.

Hugo Aimar, Ivana Gomez. Anity and distance. On the Newtonian structure of some data

kernels. Anal. Geom. Metr. Spaces. 2018;6:89-95.

Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P. Geometric deep learning: going

beyond euclidean data. IEEE Signal Processing Magazine. 2017;34(4):18-42.