Related papers: Reeb Graph of Sample Thickenings
We study the question of approximating a compact geodesic metric space by metric graphs satisfying a uniform upper bound on their first Betti number. We prove that, up to a suitable multiplicative constant, Reeb graphs of distance functions…
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques…
Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar…
Characterizing motif (i.e., locally connected subgraph patterns) statistics is important for understanding complex networks such as online social networks and communication networks. Previous work made the strong assumption that the graph…
Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the…
In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary…
Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are…
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we…
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra…
The present paper is on a new result on so-called realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages. The Reeb graph of a smooth function is a graph obtained by identifying two points in the manifold…
For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in…
There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on…
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…
Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes a metric graph and can therefore be used as a…
The problem of computing topological distance between two scalar fields based on Reeb graphs or contour trees has been studied and applied successfully to various problems in topological shape matching, data analysis, and visualization.…
In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main…