Nihat Ay, Nils Bertschinger, Jürgen Jost, Eckehard Olbrich

Paper #: 06-08-028

We develop a unifying approach for complexity measures, 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. 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.