Gaussian processes classify graphs using vertex and edge features.
problem Graph classification in machine learning.
method Transform graph features into spectral Euclidean features, apply Hodge decomposition.
result Gaussian processes can classify graphs using vertex and edge features.
Graph cross network improves graph classification accuracy.
problem Improving graph classification accuracy.
method Graph cross network (GXN) with vertex infomax pooling (VIPool) and feature-crossing layer.
result Improves graph classification accuracy by 2.12% and 1.15%.
Paper studies vertex correspondence recovery in correlated graphs with node features.
problem Recovering hidden vertex correspondence between two correlated graphs with observed edge weights and node features.
method Introduced featured correlated Gaussian Wigner model and proposed QPAlign algorithm for quadratic programming relaxation.
result Characterized optimal information-theoretic thresholds for exact and partial recovery of latent mapping.
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…
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…
Graph matching with feature vectors is solved using a two-layer graph neural network.
problem Graph matching in the presence of sparse binary features.
method Two-layer graph neural network with graph structure.
result Graph neural network can recover correct mapping with high probability under certain conditions.
Pyramidal GNN combines RC and pooling for efficient graph embeddings.
problem Efficiently embedding graphs while maintaining accuracy.
method Alternates RC layers with pooling to reduce complexity.
result Formally shows how pooling reduces complexity and speeds convergence.
The paper explores how to find relevant vertices in one graph using another graph's attributes and structure.
problem Finding relevant vertices in one graph using another graph's attributes and structure.
method Theoretical and practical exploration of vertex nomination schemes that leverage both content (edge and vertex attributes) and context (network topology).
result Necessary and sufficient conditions for schemes that use both content and context to outperform those using only one.
DeepMap learns deep graph representations via CNNs, improving graph classification performance.
problem Quantifying graph similarities for tasks like classification.
method Proposes DeepMap framework extending CNNs to arbitrary graphs, learning dense low-dimensional vectors.
result DeepMap achieves state-of-the-art performance on graph classification benchmarks.
Graph neural networks often assume vertex labels are independent, but we show this is rarely true and propose a method to improve predictions.
problem Graph neural networks often assume vertex labels are conditionally independent given their neighborhood features, which is rarely true.
method We model the joint distribution of residuals on vertices with a parameterized multivariate Gaussian and estimate parameters by maximizing the marginal likelihood of the observed labels.
result Our method achieves substantially higher accuracy than competing baselines and can be interpreted as the strength of correlation among connected vertices.
Proposes a new method to describe graph vertex features using characteristic functions.
problem Describing the distribution of vertex features at multiple scales on graphs.
method Introduces FEATHER, a computationally efficient algorithm to calculate characteristic functions based on random walk transition probabilities.
result Demonstrates that the proposed method creates high-quality graph representations and is robust to data corruption.
This paper examines how graph topology affects adversarial attacks on vertex classification.
problem Adversarial attacks on vertex classification are vulnerable to graph topology changes.
method Examined two topological graph characteristics and their impact on adversary perturbation budgets.
result Training sets including high-degree vertices or those ensuring all unlabeled nodes have neighbors can significantly increase the adversary's perturbation budget.
Vertex distortion detects if a knot is unknot.
problem Determining if a knot is the unknot.
method Using Denne-Sullivan's bound on Gromov distortion, the vertex distortion of nontrivial lattice knots is bounded. Then, it is shown that trivial vertex distortion implies the unknot.
result The conjecture that trivial vertex distortion implies the unknot is proven.
Given a time series of graphs G(t) = (V, E(t)), t = 1, 2, ..., where the fixed vertex set V represents "actors" and an edge between vertex u and vertex v at time t (uv \in E(t)) represents the existence of a communications event between actors u and v during the tth time period, we wish to detect anomalies and/or chang…
New algorithms SVCA and SSPA improve robustness to noise in nonnegative matrix factorization.
problem Estimating vertices from noisy data points in convex hull.
method Smoothed VCA (SVCA) and Smoothed SPA (SSPA) algorithms.
result Improved robustness to noise compared to existing methods.
The study finds the bounds of vertex orbits in maps derived from specific lattices.
problem Determining the bounds of vertex orbits in maps derived from k-vertex-homogeneous lattices. method Analyzing maps as quotients of k-vertex-homogeneous lattices. result Sharp bounds of the number of vertex orbits are identified.
A piecewise flat Finsler metric on a triangulated surface M is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
Given a vertex of interest in a network G1, the vertex nomination problem seeks to find the corresponding vertex of interest (if it exists) in a second network G2. A vertex nomination scheme produces a list of the vertices in G2, ranked according to how likely they are judged to be the corresponding vertex of …
634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.
problem Constructing and classifying triangulations of the octonionic projective plane.
method Combinatorial construction and analysis of symmetry groups.
result Found 634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations.
Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Network embedding methodologies, which learn a distributed vector representation for each vertex in a network, have attracted considerable interest in recent years. Existing works have demonstrated that vertex representation learned through an embedding method provides superior performance in many real-world applicatio…
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
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 [p3,3] for p≡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.
problem Investigating vertices in curves with singular points in the Euclidean plane.
method Defining vertices using evolutes of frontals and analyzing conditions for the four vertex theorem.
result Conditions for the four vertex theorem to hold for closed frontals.
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
Graph neural networks suffer from oversmoothing, but adding residual connections helps.
problem Oversmoothing in deep graph neural networks where features become indistinguishable.
method Analyzed asymptotic oversmoothing rates with and without residual connections using the multiplicative ergodic theorem.
result Adding residual connections effectively mitigates or prevents oversmoothing.
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…
In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function κ, provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classi…
We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.
Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…
A new method for fast graph embedding using diffusion graphs.
problem Efficiently generating graph embeddings for large networks.
method Diffusion graphs for rapid vertex sequence generation.
result Improved accuracy and performance with higher edge density.
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connec…
Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
Solves Skopenkov's problem on graph embedding criteria.
problem Criteria for toroidal embedding of one-vertex ribbon graphs.
method Analyzes one-vertex ribbon graphs with additional disc structure.
result Provides solutions to Skopenkov's problem.
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.
problem Vertex embedding and community detection.
method Normalized one-hot graph encoder and rank-based cluster size measure.
result Excellent numerical performance of graph encoder ensemble algorithm.
New invariants distinguish spatial graphs not previously possible.
problem Distinguishing spatial graphs using Dehn colorings.
method Developed vertex-weight invariants based on Dehn colorings.
result Found spatial graphs distinguishable by vertex-weight invariants.
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.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.
problem Understanding the geometry of smooth surfaces in 3D space.
method Analyzes the vertex curve, related to differential geometry and symmetry sets of isophote curves.
result Establishes connections between the vertex curve and other geometric curves like parabolic and flecnodal curves.
Let M be a Riemannian manifold. For p∈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.
problem Finding the root vertex in large growing networks.
method Constructs confidence sets for the root vertex in various random network models.
result Confidence sets of size independent of the number of vertices contain the root vertex with high probability.
A new graph kernel uses LCS and Wasserstein distance for better graph comparisons.
problem Graph learning methods can be limited by information from distant vertices and path length constraints.
method Proposes a Graph Kernel based on LCS similarity and Wasserstein distance in a novel metric space.
result The new kernel emphasizes comparisons between similar paths and reduces information loss.
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in R4; 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 A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A. We show vertex finiteness when G…
Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.