Related papers: Quantifying force networks in particulate systems
Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…
We study the geometry of forces in some simple models for granular stackings. The information contained in geometry is complementary to that in the distribution of forces in a single inter-particle contact, which is more widely studied. We…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Force networks form the skeleton of static granular matter. They are the key ingredient to mechanical properties, such as stability, elasticity and sound transmission, which are of utmost importance for civil engineering and industrial…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small…
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 models. Deeply rooted in algebraic topology, persistent…
The force networks of different granular ensembles are defined and their topological properties studied using the tools of complex networks. In particular, for each set of grains compressed in a square box, it is introduced a force…
As a granular material is compressed, the particles and forces within the system arrange to form complex heterogeneous structures. Force chains are a prime example and are thought to constrain bulk properties such as mechanical stability…
In the companion paper~\cite{paper1}, we use classical measures based on force probability density functions (PDFs), as well as Betti numbers (quantifying the number of components, related to force chains, and loops), to describe the force…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…
Force-directed layouts belong to a popular class of methods used to position nodes in a node-link diagram. However, they typically lack direct consideration of global structures, which can result in visual clutter and the overlap of…
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…
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 of persistent homology from scratch using some examples. We…
We study the relation between the persistent homology and the spectral sequence of a filtered chain complex over a field. Our method is based on a decomposition of the persistent homology. We demonstrate that, under fairly general…