Self-dissimilarity as a high dimensional complexity measure

For many systems characterized as "complex" the patterns exhibited on different scales differ markedly from one another. For example the biomass distribution in a human body "looks very different" depending on the scale at which one examines it. Conversely, the patterns at different scales in "simple" systems (e.g., gases, mountains, crystals) vary little from one scale to another. Accordingly, the degrees of self-dissimilarity between the patterns of a system at various scales constitute a complexity "signature" of that system.

Here I present a novel quantification of self-dissimilarity. This quantification can be measured for many kinds of real-world data. This allows comparisons of the complexity signatures of wholly different kinds of systems (e.g., systems involving information density in a digital computer vs. species densities in a rain-forest vs. capital density in an economy, etc.). Moreover, in contrast to many other suggested complexity measures, evaluating the self-dissimilarity of a system does not require one to already have a model of the system.

These facts may allow self-dissimilarity signatures to be used as the underlying observational variables of an eventual overarching theory relating all complex systems. To illustrate self-dissimilarity I present several numerical experiments. In particular, I show that underlying structure of the logistic map is picked out by the self-dissimilarity signature of time series' produced by that map.