Approccio Strutturale e Decomposizione di Complessi Simpliciali

Knowledge extraction and interpretation from high-dimensional, incomplete and noisy datasets is an important current challenge as is the necessity to model the complex systems from which such data usually originates. Topological Data Analysis provides a general framework to study the shape of data by leveraging algebraic topology and other tools from pure mathematics. However it has been used mainly as a qualitative method, the problem beeing the lack of proper tools to perform effective statistical analysis. In this context, we focused our study on the local structure of abstract simplicial complexes. The rationale is that, differently from well known complex networks, simplicial complexes encode also non dyadic relationships. In particular, we defined Simplicial Degrees Matrices associated to maximal simplices. They can be seen as new lenses from which to analyze interactions within complex systems. For this reason, we first studied their structure on synthetic models of complex networks, then we defined some metrics on them in order to measure the local and non local shape of data. Finally we leveraged the obtained insights to provide interpretations of real datasets.