Approximate maximizers of intricacy functionals

TL;DR

This paper precisely calculates the growth rate of neural complexity functionals by constructing approximate maximizers, revealing properties of subsystems and entropy profiles in complex systems.

Contribution

It introduces a method to compute the growth speed of intricacy functionals and constructs approximate maximizers applicable across all intricacies.

Findings

Exact growth speed of intricacy functionals determined

Existence of a subsystem size threshold in approximate maximizers

Most small subsystems are nearly uniform, larger ones determine the whole system

Abstract

G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the neural complexity of a family of random variables. This functional is a special case of intricacy, i.e., an average of the mutual information of subsystems whose weights have good mathematical properties. Moreover, its maximum value grows at a definite speed with the size of the system. In this work, we compute exactly this speed of growth by building "approximate maximizers" subject to an entropy condition. These approximate maximizers work simultaneously for all intricacies. We also establish some properties of arbitrary approximate maximizers, in particular the existence of a threshold in the size of subsystems of approximate maximizers: most smaller subsystems are almost equidistributed, most larger subsystems determine the full system. The main ideas are a random construction of almost maximizers with a high statistical symmetry and the consideration of entropy profiles, i.e., the average entropies of sub-systems of a given size. The latter gives rise to interesting questions of probability and information theory.