Networks, Markov Lie Monoids, and Generalized Entropy

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 non-negative values, one obtains exactly all Markov transformations in n dimensions that are continuously connected to the identity. Networks are defined by a set of n nodes (points) along with the connections among some pairs of nodes. Such a network can be represented by a connection (or connectivity, or adjancy) matrix Cij whose off-diagonal elements give the non-negative ‘strength’ of the connection between nodes i and j in the network. 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 th...

• Type: Conference
• Year: 2005
• Where: MMMACNS
• Authors: Joseph E. Johnson
• Keywords: Computer Networks, Generalized Entropies, Lie Groups, Markov Transformations, MMMACNS 2005