Dynamical distance: coarse grains, pattern recognition, and network analysis

A system moves randomly in a space of states X governed by a stochastic matrix R. From the dynamics alone a distance function is defined on X. This distance allows a coarse graining that is optimal with respect to the dynamics in the following sense: if all grains are of diameter less than ɛ then the coarse grained dynamics and original dynamics differ by less than ɛ, which is to say the coarse graining is commutative up to that level. Using this distance function two applications are considered: the creation of “words” or concepts in pattern recognition and the identification of communities in networks.