G. Edelman, O. Sporns, and G. Tononi have introduced the neural complexity of
a family of random variables, defining it as a specific average of mutual
information over subfamilies. We show that their choice of weights satisfies
two natural properties, namely exchangeability and additivity, and we call any
functional satisfying these two properties an intricacy. We classify all
intricacies in terms of probability laws on the unit interval and study the
growth rate of maximal intricacies when the size of the system goes to
infinity. For systems of a fixed size, we show that maximizers have small
support and exchangeable systems have small intricacy. In particular,
maximizing intricacy leads to spontaneous symmetry breaking and failure of
uniqueness.Comment: minor edit
