The following text from the website of the Complex Systems Society addresses the question "What are Complex Systems?" and perfectly describes the main concept that underlies this project: "The most famous quote about Complex Systems comes from Aristole who said that 'The whole is more than the sum of its parts'. Complex systems are systems where the collective behavior of their parts entails emergence of properties that can hardly, if not at all, be inferred from properties of the parts."
We propose a geometric formalization of this concept. The complexity of a system is quantified as its deviation from the sum of its parts which is interpreted as a geometric projection. While our initial approach was based on information geometry only, the current study also applies the theory of hierarchical and, in particular, graphical models and causality theory based on Bayesian networks. This allows for an integrated analysis of the interplay of causal interactions, stochastic dependence, and complexity.
Relations to and among other approaches to complexity are studied. We are particularly interested in understanding how algorithmic notions of complexity correspond to probabilistic ones, similar to the wellknown close connection between algorithmic complexity and Shannon entropy. In that context, various complexity measures for stochastic processes are related to corresponding complexities of typical process realizations, thereby identifying similarities of seemingly different concepts.
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