Behavior of information-content variance in high-dimensional and multiscale systems

Characterize the behavior of the variance-of-information-content complexity measure C in high-dimensional and multiscale systems, particularly with respect to coarse graining and symbolic representations.

Background

The paper proposes the variance of the information content, denoted by C, as a statistical complexity measure. For a probability distribution with state probabilities p_alpha, C is defined as the variance of the information content, C = sum_alpha p_alpha (ln p_alpha)^2 - S^2, where S is the Shannon entropy. The authors demonstrate its properties for finite-state distributions, Boltzmann–Gibbs systems, the two-dimensional Ising model, chaotic maps, and fractional Gaussian noise.

The paper does not resolve how C behaves when applied to systems with many degrees of freedom or multiple characteristic scales. In particular, it leaves unresolved how coarse-graining procedures and symbolic representations affect the measure in such settings, motivating further characterization of its applicability and interpretation beyond the examples studied.