Networks, Markov Lie Monoids, and Generalized Entropy

  • Joseph E. Johnson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3685)


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.


Generalize Entropy General Linear Group Eigenvalue Spectrum Connection Matrix Renyi Entropy 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Joseph E. Johnson
    • 1
  1. 1.Department of PhysicsUniversity of South CarolinaColumbia

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