Applications of Category Theory in Analysis of Complex Systems

Complex systems are ubiquitous in many domains ranging from social sciences and biology to distributed systems and software engineering. A system is classified as complex when its properties cannot be expressed as an aggregation of the properties of its components. Since such a distinction is qualitative, complex systems cannot be analysed adequately using methodology developed for simple systems. In this thesis we develop foundations of a categorical framework with the goal of facilitating design and analysis of complex systems based on macro-level specifications and providing guidelines for controlled transitions to desired target systems under a given set of constraints. Our formalisation is based on algebraic theory of Graph Grammars (GGs) which is essentially a rule-based mechanism for generating, manipulating, and analysing graphs. The behaviour of a GG can be captured comprehensively and efficiently using a process called unfolding. However, unfolding theory, as originally proposed, is not sufficiently expressive to allow for data attributes or preconditions forbidding application of a rule if certain nodes or edges are present. Since many complex systems have these features, we extend the theory of unfolding of GGs in three directions (namely conditional, attributed, and conditional attributed) in order to provide sufficient expressiveness for modelling and analysing complex systems. We also show how negative reconditions can be encoded using attributes and complementation. At the end, we apply the conditional attributed unfolding within a case study to analyse power dynamics in socio-political systems modelled as graph grammars.