Networks, Markov Lie Monoids, and Generalized Entropy

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.