A Review of Complexity Measures from Epiplexity to Algorithmic Information Dynamics

Complexity science has proposed many measures of complexity, structure, information, emergence and (self-)organisation, often in isolation and without systematic understanding of their relationships, redundancies or distinct properties. As new measures continue to appear, comparative assessment is increasingly needed. Here, we systematically compare classical and recent complexity measures, characterising their relationships and sensitivities in discriminating the qualitative behaviour of discrete and continuous dynamical systems. The primary analysis used an initial 11-system benchmark comprising four elementary cellular automata, Lorenz dynamics, Mandelbrot and Julia fractals, three Boolean-network topologies, and a yeast association network. Across the binarization threshold sweep, Block-level Algorithmic Information Dynamics (AID) (\( |\Delta\mathrm{BDM}| \)) preserved cross-system rankings with mean Spearman agreement 0.881 (95\% CI, 0.848--0.910), \( \mathrm{ICC}(3,1)=0.893 \), and Kendall's \( W=0.894 \). Independent validation comprised a five-regime ladder spanning trivial/ordered, periodic/ordered, critical, chaotic, and maximally random dynamics, together with a continuous logistic-map sweep. In the categorical analysis, 2-D cell and block AID ranked first and second, respectively, in the task-specific composite combining categorical discrimination, Lyapunov tracking, and assigned perturbation capability. Post-hoc, 1-D BDM and Epiplexity separated critical from chaotic dynamics. Epiplexity and expiplexity-derived coding statistics did not distinguish chaotic from maximally random endpoints, whereas AID additionally retained signed, localised, multiscale intervention effects. Sequential structural epiplexity \( S_T \) separated validation categories (\( \epsilon_H^2=0.626 \), 95\% interval 0.515--0.782), with a much lower point separation effect size than Cell and Block AID (BDM perturbation). We develop a proposed characterisation of Epiplexity as a functional over an AID information landscape, making explicit the sufficiency assumptions needed to identify it with the original bounded-MDL definition. Thus, CTM/BDM provide estimators of algorithmic information, AID provides the perturbational calculus, and explicitly constructed scalar observables project selected aspects of that landscape.