arXiv research

New proof for global rigidity of vertex scaling on polyhedral surfaces.

Network embedding aims to learn the low-dimensional representations of vertexes in a network, while structure and inherent properties of the network is preserved. Existing network embedding works primarily focus on preserving the microscopic structure, such as the first- and second-order proximity of vertexes, while th…

Graph cross network improves graph classification accuracy.

In this paper, we develop a new aligned vertex convolutional network model to learn multi-scale local-level vertex features for graph classification. Our idea is to transform the graphs of arbitrary sizes into fixed-sized aligned vertex grid structures, and define a new vertex convolution operation by adopting a set of…

Method analyzes large-scale network data to detect communication pattern shifts.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

We focus on developing a novel scalable graph-based semi-supervised learning (SSL) method for a small number of labeled data and a large amount of unlabeled data. Due to the lack of labeled data and the availability of large-scale unlabeled data, existing SSL methods usually encounter either suboptimal performance beca…

Proves rigidity of circle packings in the plane, generalizing previous work.

Efficiently projects points onto polytopes, especially useful in web-scale applications.

Fractional combinatorial flow improves surface conformal structures.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

Vertex distortion detects if a knot is unknot.

The study finds the bounds of vertex orbits in maps derived from specific lattices.

We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < ε(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $ε(x_i)$ varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whethe…

Given a vertex of interest in a network $G_1$, the vertex nomination problem seeks to find the corresponding vertex of interest (if it exists) in a second network $G_2$. A vertex nomination scheme produces a list of the vertices in $G_2$, ranked according to how likely they are judged to be the corresponding vertex of …

Graph neural networks often assume vertex labels are independent, but we show this is rarely true and propose a method to improve predictions.

Vertex centrality measures are a multi-purpose analysis tool, commonly used in many application environments to retrieve information and unveil knowledge from the graphs and network structural properties. However, the algorithms of such metrics are expensive in terms of computational resources when running real-time ap…

634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.

This paper shows semi-equivelar toroidal maps are vertex-transitive covers.

Defines formal vertex laws related to Lie conformal algebras.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particu…

The study examines vertices in curves with singular points in the Euclidean plane.

Vertex distortion measures how far lattice knots deviate from straight lines.

For random graphs distributed according to stochastic blockmodels, a special case of latent position graphs, adjacency spectral embedding followed by appropriate vertex classification is asymptotically Bayes optimal; but this approach requires knowledge of and critically depends on the model dimension. In this paper, w…

Proposes a new method to describe graph vertex features using characteristic functions.

We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.

Proves a generalized Whitehead cut vertex lemma for tree groups.

Solves Skopenkov's problem on graph embedding criteria.

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, …

Efficient method for vertex embedding and community detection.

We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $ρ$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $ρ$ is decomposed into connected components. We show that the vertex-partition induced by the thresh…

New invariants distinguish spatial graphs not previously possible.

Given a graph in which a few vertices are deemed interesting a priori, the vertex nomination task is to order the remaining vertices into a nomination list such that there is a concentration of interesting vertices at the top of the list. Previous work has yielded several approaches to this problem, with theoretical re…

The paper explores how to find relevant vertices in one graph using another graph's attributes and structure.

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a…

Proofs for Moon's theorem and its generalization.

Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …

Study finds root vertex in large networks with high probability.

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we e…

Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G…

Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…

Tree++ graph kernel captures similarities at multiple granularities.

This paper proposes a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model that can directly learn a classification function for graphs of arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network (GCNN) models, the proposed QSGCNN model incorporates the process of identifying transit…

Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are quasi-isometric if and only if every one-ended vertex group of A is quasi-isometric to…