Mathematical Foundations of Networks Supporting Cluster Identification

Joseph E. Johnson, John William Campbell

2014

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.

References

  1. Johnson, Joseph E, 1985 Markov-Type Lie Groups in GL(n,R) Journal of Mathematical Physics 26 (2) 252- 257
  2. Johnson, Joseph E. 2005 Networks, Markov Lie Monoids, and Generalized Entropy, Computer Networks Security, Third International Workshop on Mathematical Methods, Models, and Architectures for Computer Network Security, St. Petersburg, Russia, Proceedings, 129-135US
  3. Johnson, Joseph E., 2006 Markov Lie Monoid Entropies as Network Metrics MIT ICCS Conference on Networks & Complex Systems
  4. Campbell, John William, 2014, Network Analysis and Cluster Detection Using Markov Theory M.S. Thesis, University of South Carolina
  5. Johnson, Joseph E. 2012 Methods and Systems for Determining Entropy Metrics for Networks US Patent 8271412.
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Paper Citation


in Harvard Style

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


in Bibtex Style

@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},
}


in EndNote Style

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