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 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. Generalized entropy can be defined on these probability conserving flows, and thus the generalized entropies for these Markov transformations become metrics for the topology of the corresponding network. In this framework one is able to carry analogies across multiple mathematical areas by applying Lie groups and algebras, Markov transformations, conserved and non-conserved diffusion flows, and generalized entropies, on the one hand, to network theory and topology, on the other. We are specifically interested in using these generalized entropies as metrics for the tracking of network topological changes such as one would expect under attacks and intrusions on internets.