ANALYSIS AND GENERATION OF DATA WITH TOPOLOGY FROM COMBINATORIAL REPRESENTATIONS

Abstract

Simplicial complexes and hypergraphs have been applied in a variety of domains for modelling data with polyadic relations and analysing global topological properties of data.

There are three complementary aspects of understanding data with topological properties. The first aspect involves analysing the data qualitatively by visualising its representation. The second aspect involves analysing the data quantitatively by computing its topological properties. The third and last aspect involves generating synthetic data with a prescribed topological constraint for simulation-based studies. Motivated by the qualitative, quantitative and data-generation aspects of analysing data with topological properties, this thesis makes three contributions.

First, we propose a family of algorithms for visualising hypergraphs. We propose metrics for good hypergraph drawing.

Second, we adopt the notion of ϵ-net to approximately compute topological features induced by the metric data. We derive several properties of ϵ-net of graphs and point clouds. We theoretically show that ϵ-nets facilitate to construct a sparse simplicial representation of data and further compute its topological features.

Finally, we propose algorithms for the construction and random generation of hypergraphs with prescribed degree and dimension sequences. We use our random generation algorithm to propose a Self-Normalised Importance Sampling estimator for estimating hypergraph properties.