Related papers: Towards Homological Methods in Graphic Statics
We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second…
Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all "Geometry of Interaction" (GoI) constructions introduced so far. This series of work was inspired from Girard's…
We carry out the homogenization of time-harmonic Maxwell's equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting,…
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
A visualized graph is a powerful tool for data analysis and synthesis tasks. In this case, the task of visualization constitutes not only in displaying vertices and edges according to the graph representation, but also in ensuring that the…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We…
We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges,…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks…
Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges…
This paper clarifies the main research methods and ideas of the thesis [1,2,4]. The special calculation process is also realized by corresponding computer algorithm. Finally, we introduce zero rows sum case and give the corresponding…
Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting…
A graph embedding algorithm embeds a graph into a low-dimensional space such that the embedding preserves the inherent properties of the graph. While graph embedding is fundamentally related to graph visualization, prior work did not…
In this article we establish some results that allow to deduce the continuity of homomorphisms of (topological) abelian groups from commutative diagrams. In particular, we present a new topological version of the classical Five-Lemma. These…
Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a…
The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with…
Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a…
We argue that the mathematical structure, enabling certain cascading and emergent phenomena to intuitively emerge, coincides with Galois connections. We introduce the notion of generative effects to formally capture such phenomena. We…