  In the past three decades, many theoretical measures of complexity have been
proposed to help understand complex systems. In this work, for the first time,
we place these measures on a level playing field, to explore the qualitative
similarities and differences between them, and their shortcomings.
Specifically, using the Boltzmann machine architecture (a fully connected
recurrent neural network) with uniformly distributed weights as our model of
study, we numerically measure how complexity changes as a function of network
dynamics and network parameters. We apply an extension of one such
information-theoretic measure of complexity to understand incremental Hebbian
learning in Hopfield networks, a fully recurrent architecture model of
autoassociative memory. In the course of Hebbian learning, the total
information flow reflects a natural upward trend in complexity as the network
attempts to learn more and more patterns.
