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A geometric approach to complexity
Nihat Ay1, Eckehard Olbrich, Nils Bertschinger
1Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany. nay@mis.mpg.de
Abstract:
We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of an interaction hierarchy. In mathematical terms, we present a theory of complexity measures for finite random fields using the geometric framework of hierarchies of exponential families. Within our framework, previously proposed complexity measures find their natural place and gain a new interpretation.
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