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Networks, Markov Lie Monoids, and Generalized Entropy

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Part of the Lecture Notes in Computer Science book series (LNCCN,volume 3685)

Abstract

The continuous general linear group in n dimensions can be decomposed into two Lie groups: (1) an n(n-1) dimensional ‘Markov type’ Lie group that is defined by preserving the sum of the components of a vector, and (2) the n dimensional Abelian Lie group, A(n), of scaling transformations of the coordinates. With the restriction of the first Lie algebra parameters to nonnegative values, one obtains exactly all Markov transformations in n dimensions that are continuously connected to the identity. In this work we show that every network, as defined by its C matrix, is in one to one correspondence to one element of the Markov monoid of the same dimensionality. It follows that any network matrix, C, is the generator of a continuous Markov transformation that can be interpreted as producing an irreversible flow among the nodes of the corresponding network.

Keywords

  • Generalize Entropy
  • General Linear Group
  • Eigenvalue Spectrum
  • Connection Matrix
  • Renyi Entropy

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References

  1. Renyi, A.: Probability Theory. North-Holland Series in Applied Mathematics and Mechanics, 670 pages. North-Holland Pub. Co., Amsterdam (1970)

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  3. Johnson, J.E.: Markov-type Lie Groups in GL(n, R). Journal of Mathematical Physics 26, 252–257 (1985)

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  4. Gudkov, V., Johnson, J.E.: Network as a complex system: information flow analysis, 10 pages (2001) arXiv:lin.CD/0110008v1

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  5. Gudkov, V., Johnson, J.E.: Chapter 1: Multidimensional network monitoring for intrusion detection, 12 pages (2002) arXiv: cs.CR/020620v1

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© 2005 Springer-Verlag Berlin Heidelberg

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Johnson, J.E. (2005). Networks, Markov Lie Monoids, and Generalized Entropy. In: Gorodetsky, V., Kotenko, I., Skormin, V. (eds) Computer Network Security. MMM-ACNS 2005. Lecture Notes in Computer Science, vol 3685. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11560326_10

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  • DOI: https://doi.org/10.1007/11560326_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29113-8

  • Online ISBN: 978-3-540-31998-6

  • eBook Packages: Computer ScienceComputer Science (R0)