This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and...

In this paper, we define the homological Morse numbers of a filtered cell complex in terms relative homology nested filtration pieces, and derive inequalities relating these to Betti tables multi-parameter persistence modules considered filtration. Using Mayer-Vietoris spectral sequence first obtain strong weak involving above quantities, then improve achieving sharp lower bound for numbers. Fu...

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study, and encode efficiently multi-scale features being increasingly used digital image classification. The represent number of connected components, cycles, voids that describe the shape data. extracts birth death these through filtration process. lifespan can be represented using persistent diagrams (topolo...

Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a single index, and is able to capture higher order features beyond the usual clustering techniques. There are three descriptive statistics of persistent homology...

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead large inputs, and tasks this area tend be expensive. We propose two methods for chain complexes free 2-parameter persistence modules. The first method extends multi-ch...

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramot...

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs networks. Persistent path (PPH) extends the with filtration to deal asymmetry structures. However, PPH is constrained purely topological persistence cannot track homotopic shape evolution of data during filtration. To overcome limitation PPH, persistent Laplacian (PPL) introduced capture ...

We study the persistent homology of an Erdős–Rényi random clique complex filtration on n vertices. Here, each edge e appears independently at a uniform time $$p_e \in [0,1]$$ , and persistence cycle $$\sigma $$ is defined as $$p_2(\sigma ) / p_1(\sigma )$$ where $$p_1(\sigma are birth death times . show that if $$k \ge 1$$ fixed, then with high probability maximal k-cycle order $$n^{1/k(k+1)}$$

We propose methods for computing two network features with topological underpinnings: the Rips and Dowker Persistent Homology Diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to intrinsic asymmetry in the data, and investigate the stability properties of both the ...

We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to two-sided filtrations. In addition to describing the theoretical background, we prese...