Network-based modeling of complex systems and data using the language of graphs has
become an essential topic across a range of different disciplines. Arguably, this graph-based …

A great success of solid state physics comes from the characterization of crystal structures in
the reciprocal (wave vector) space. The power of structural characterization in Fourier space …

" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

Persistent homology (PH) is a method used in topological data analysis (TDA) to study
qualitative features of data that persist across multiple scales. It is robust to perturbations of …

The continued and dramatic rise in the size of data sets has meant that new methods are
required to model and analyze them. This timely account introduces topological data …

A suitable feature representation that can both preserve the data intrinsic information and
reduce data complexity and dimensionality is key to the performance of machine learning …

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they
are routinely used to describe succinctly complex topological properties of complicated …

In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

The study of point defects in non-metallic crystals has become relevant for an increasing
number of materials applications. Progress requires a foundation of consistent definitions …

This paper introduces persistent homology, which is a powerful tool to characterize the
shape of data using the mathematical concept of topology. We explain the fundamental idea …