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A Geometric Approach to Complexity
Nihat Ay, Eckehard Olbrich, Nils Bertschinger and Jürgen Jost
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.