Inference of Hierarchical Structure in Complex Systems

Doctoral thesis, 2008

Hierarchical organization is a central property of complex systems. It is even argued that a system is required to be hierarchical in order to evolve complexity within reasonable time. A hierarchy of a system is defined as the set of self-contained levels at which the system operates and can be described on. Given a dynamical system there are only specific levels that are valid. This thesis mainly concerns the definition and inference of such levels. Paper I describes an algorithm for finding hierarchical levels in stochastic processes. The method systematically explores the set of possible partitions of a process' state space and statistically determines which of the partitions that impose closed dynamics. It is applicable to moderately sized systems. In Paper II an alternative approach that applies to linear dynamical systems is presented. In this case the spectral properties of the matrix that defines a system's dynamics is utilized, which allows for analysis of large systems (with on the order of thousand states). The specification and analysis of an algorithm that is based on the results in Paper II is presented in Paper III. Paper IV applies the spectral method and a complementary agglomeration method to infer aggregated dynamics in a Markov model of codon substitutions in DNA. The standard genetic code is identified as a projection that gives the hierarchical level of amino acid substitutions. Further, higher order amino acid groups that are relatively conserved under substitutions are found to define other levels of dynamics. Paper V and VI relate hierarchical organization to primordial evolution in a conceptual model that is based on the RNA world hypothesis. A well-stirred system of processes that catalyze the production of other processes is shown to successively build higher levels of organization from simple and general-purpose components by autocatalysis.