Related papers: Measure Theoretic Reeb Graphs and Reeb Spaces
In a graph $A$, the measure $|M_g^A(f)|=m_g^A(f)$ for each arbitrary edge $f=gh$ counts the edges in $A$ closer to $g$ than $h$. $A$ is termed an edge quasi-$\lambda$-distance-balanced graph in a metric space (abbreviated as $EQDBG$), where…
As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental,…
In this paper, we define several measures induced by a finite directed graph. The study themselves is interesting ont only in the noncommutative probability point of view but also in the algebraic structure point of view, since to define…
The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…
Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…
Persistence diagrams (PDs), often characterized as sets of death and birth of homology class, have been known for providing a topological representation of a graph structure, which is often useful in machine learning tasks. Prior works rely…
Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by…
Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an…
Real-world graphs have inherently complex and diverse topological patterns, known as topological heterogeneity. Most existing works learn graph representation in a single constant curvature space that is insufficient to match the complex…
In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating…
Temporal graphs are commonly used to represent time-resolved relations between entities in many natural and artificial systems. Many techniques were devised to investigate the evolution of temporal graphs by comparing their state at…
We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…
Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…
Robustness is a critical measure of the resilience of large networked systems, such as transportation and communication networks. Most prior works focus on the global robustness of a given graph at large, e.g., by measuring its overall…
The rankability of data is a recently proposed problem that considers the ability of a dataset, represented as a graph, to produce a meaningful ranking of the items it contains. To study this concept, a number of rankability measures have…
The rise of graph analytic systems has created a need for ways to measure and compare the capabilities of these systems. Graph analytics present unique scalability difficulties. The machine learning, high performance computing, and visual…
Topological metrics of graphs provide a natural way to describe the prominent features of various types of networks. Graph metrics describe the structure and interplay of graph edges and have found applications in many scientific fields. In…
We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data…