A Geometric Approach to Complexity

Nihat Ay, Eckehard Olbrich, Nils Bertschinger, and Jürgen Jost

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Submission date: 24. Aug. 2011
Pages: 22
published in: Chaos, 21 (2011) 3, art-no. 037103 
DOI number (of the published article): 10.1063/1.3638446
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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.