Self-Dissimilarity: An Empirically Observable Complexity Measure

For systems usually characterized as \complex/living/intelligent" very often the spatio-temporal patterns exhibited on di erent scales di er markedly from one another. For example the biomass distribution of a human body \looks very di erent" depending on the spatial scale at which one examines that biomass. Conversely, the density patterns at di erent scales in \dead/simple" systems (e.g., gases, mountains, crystals) do not vary signi cantly from one another. Accordingly, the degrees of self-dissimilarity between various scales constitutes a complexity \signature" of the system. Such signatures can be empirically measured for many real-world data sets concerning spatio-temporal densities, be they mass densities, species densities, or symbol densities. This allows us to compare the complexity signatures of wholly di erent kinds of systems (e.g., systems involving information density in a digital computer, vs. species densities in a rainforest, vs. capital density in an economy, etc.). Such signatures can also be clustered, to provide an empirically determined taxonomy of \kinds of systems" that share organizational traits. The precise measure of dissimilarity between scales that we propose is the amount of extra information on one scale beyond that which exists on a di erent scale. This \added information" is perhaps most naturally determined using a maximum entropy inference of the distribution of patterns at the second scale, based on the provided distribution at the rst scale. We brie y discuss using our measure with other inference mechanisms (e.g., Kolmogorov complexity-based inference).