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Paper

Mathematical Foundations of Networks Supporting Cluster IdentificationTopics: Clustering and Classification Methods; Data Analytics; Mining High-Dimensional Data; Web Mining

Keyword(s):Cluster Identification, Network Series Expansions, Renyi Entropy Spectra, Markov Type Lie Group, Continuous Groups, Lie Group.

Related
Ontology
Subjects/Areas/Topics:Artificial Intelligence
;
Business Analytics
;
Clustering and Classification Methods
;
Data Analytics
;
Data Engineering
;
Knowledge Discovery and Information Retrieval
;
Knowledge-Based Systems
;
Mining High-Dimensional Data
;
Soft Computing
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Symbolic Systems
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Web Mining

Abstract: The author proved that the continuous general linear (Lie) group in n dimensions can be decomposed into (a) a Markov type Lie group (MTLG) preserving the sum of the components of a vector, and (b) an Abelian Lie scaling group that scales each of the components. For a specific Lie basis, the MTLG generated all continuous Markov transformations (a Lie Markov Monoid LMM) and in subsequently published work, proved that every possible network as defined by an n x n connection matrix Cij of non-negative off-diagonal real numbers was isomorphic to the set of LMM. As this defined the diagonal of C, it supported full eigenvalue analysis of the generated Markov Matrix as well as support of Renyi entropies whose spectra ordered the nodes and make comparison of networks now possible. Our new research provides (a) a method of expanding a network topology in different orders of Renyi entropies, (b) the construction of a meta-network of all possible networks of use in network classification, (c) the use of eigenvector analysis of the LMM generated by a network C to provide an agnostic methodology for identifying clusters and (d) an a methodology for identifying clusters in general numeric database tables.(More)

The author proved that the continuous general linear (Lie) group in n dimensions can be decomposed into (a) a Markov type Lie group (MTLG) preserving the sum of the components of a vector, and (b) an Abelian Lie scaling group that scales each of the components. For a specific Lie basis, the MTLG generated all continuous Markov transformations (a Lie Markov Monoid LMM) and in subsequently published work, proved that every possible network as defined by an n x n connection matrix Cij of non-negative off-diagonal real numbers was isomorphic to the set of LMM. As this defined the diagonal of C, it supported full eigenvalue analysis of the generated Markov Matrix as well as support of Renyi entropies whose spectra ordered the nodes and make comparison of networks now possible. Our new research provides (a) a method of expanding a network topology in different orders of Renyi entropies, (b) the construction of a meta-network of all possible networks of use in network classification, (c) the use of eigenvector analysis of the LMM generated by a network C to provide an agnostic methodology for identifying clusters and (d) an a methodology for identifying clusters in general numeric database tables.

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E. Johnson, J. and William Campbell, J. (2014). Mathematical Foundations of Networks Supporting Cluster Identification.In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval - Volume 1: KDIR, (IC3K 2014) ISBN 978-989-758-048-2, pages 277-285. DOI: 10.5220/0005086902770285

@conference{kdir14, author={Joseph E. Johnson. and John William Campbell.}, title={Mathematical Foundations of Networks Supporting Cluster Identification}, booktitle={Proceedings of the International Conference on Knowledge Discovery and Information Retrieval - Volume 1: KDIR, (IC3K 2014)}, year={2014}, pages={277-285}, publisher={SciTePress}, organization={INSTICC}, doi={10.5220/0005086902770285}, isbn={978-989-758-048-2}, }

TY - CONF

JO - Proceedings of the International Conference on Knowledge Discovery and Information Retrieval - Volume 1: KDIR, (IC3K 2014) TI - Mathematical Foundations of Networks Supporting Cluster Identification SN - 978-989-758-048-2 AU - E. Johnson, J. AU - William Campbell, J. PY - 2014 SP - 277 EP - 285 DO - 10.5220/0005086902770285