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Self-Dissimilarity: An Empirical Complexity Signature

Presenter: William Macready
When: April 25, 2005 11AM PDT

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 is very different depending on
the scale at which one examines it. Conversely, the patterns at
different scales in “simple’' systems (e.g., gases, crystals) vary
little from one scale to another. Accordingly, the degrees of
self-*dis*similarity between the patterns of a system at various
scales constitute a complexity “signature’' of that system.

Here we present a quantification of self-dissimilarity that
can be measured for many kinds of real-world data. This allows
comparisons of the complexity signatures of wholly different kinds of
spatio-temporal patterns, e.g., that of amino acid populations within
a Eukaryote, cell-type distributions in a multi-cellular organization,
information density in a digital computer, species densities in a
rain-forest, capital density in an economy, etc.

In addition to its breadth of applicability,
self-dissimilarity does not require one to already have a model of the
system before being able to measure it. Accordingly self-
dissimilarity may serve as an important observational variable of an
eventual overarching theory relating all complex / “living” systems.

To illustrate self-dissimilarity we present several numerical
experiments. In particular, we show that the underlying structure of the
logistic map is picked out by the self-dissimilarity signature of time
series’ produced by that map.

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