"Self-dissimilarity as a high dimensional complexity measure"
For essentially any system commonly characterized as "complex", the spatio-temporal patterns exhibited on different scales differ markedly from one another. Biological organisms obviously exhibit this nature. The Earth climate system is another excellent example, having very different dynamic processes operating at all spatiotemporal scales. Complex human artifacts also share this property, as anyone familiar with large-scale engineering projects will attest. Conversely, the patterns at different scales in "simple" systems (e.g., gases, crystals) do not vary significantly from one another, and therefore allow the entire pattern over all scales to be encoded into a short description. It is the self-similar aspects of such systems, as revealed by allometric scaling, scaling analysis of networks, etc., that reflects their inherently simple nature. Accordingly, it is the self-dissimilarity (SD) between the patterns at various scales that constitutes the complexity "signature" of a system. Intuitively, such a signature tells us how the information stored at one scale in a system and its processing at that scale is related to the information and processing at the other scales. Highly different information processing at different scales means the system is efficient at encoding as much processing into its dynamics as possible, whereas having little differences between the various scales is often associated with robust dynamics. The SD signature of a system is a function purely of the spatio-temporal pattern of that system, and does not directly depend on the rules generating that pattern. This means that while SD signatures may incorporate prior knowledge about the system generating a data set (to statistically extend that data set), they are functions of data sets rather than of models of the underling system. They therefore provide model-free synopses of the information-processing of the underlying system. This model-independence allows SD analysis to be applied to a broad range of types of systems/data, and for the associated SD signatures to be compared. This suggests that such analysis may prove quite useful in formulating a broadly applicable science of complex systems. This paper presents a formalization of SD, taking care to show how it differs from previously suggested complexity measures. We demonstrate this formalization on several examples, including the logistic map both with and without noise. Other examples are various two-dimensional patterns, including photographs, crystal patterns, and multifractals. These examples illustrate the potential utility of SD analysis for understanding complex systems.