arXiv research

Bi-directional Curriculum Learning improves graph anomaly detection by considering both homogeneity and heterogeneity.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.

In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled Seifert graph.

Hyperbolic groups' infinite orbits spread evenly in spaces.

Free groups can be end homogeneity groups of 3-manifolds.

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a part…

GHNet improves graph learning by balancing homogeneity and heterogeneity.

AEGCN uses autoencoder constraints to improve graph node classification.

No stable discrete maps into certain curved spaces exist.

This paper analyzes various graph clustering methods and their applications.

Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix…

Graph node embedding aims at learning a vector representation for all nodes given a graph. It is a central problem in many machine learning tasks (e.g., node classification, recommendation, community detection). The key problem in graph node embedding lies in how to define the dependence to neighbors. Existing approach…

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…

New insights into ends of quotient spaces and graphs.

Study on affine surfaces with specific algebraic properties.

The aim of this short note is to draw attention to a method by which the partition function and marginal probabilities for a certain class of random fields on complete graphs can be computed in polynomial time. This class includes Ising models with homogeneous pairwise potentials but arbitrary (inhomogeneous) unary pot…

New method for learning on heterogeneous graphs without meta-paths.

A new definition of umbilic points at infinity for polynomial surfaces.

Proposes methods for local clustering in attributed graphs.

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomi…

Attention-based GNNs can't prevent oversmoothing, leading to homogeneous node representations.

In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$. As a straightforward consequence, we announce a structure-type result for proper graphs defined on relatively compact domains.

The paper introduces heterogeneous manifolds for better graph embeddings.

We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic if and only if their graphs are isomorphic. We also briefly discuss some metric …

Study on minimal surfaces in a specific homogeneous space with non-existence and construction results.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

Graph neural networks improve topology control of power grids.

An algorithm finds optimal covariates for blocking in randomized experiments.

We prove that the existence of a positively defined, invariant Einstein metric $m$ on a connected homogeneous space $G/H$ of a compact Lie group $G$ is the consequence of non-contractibility of some compact set $C=X_{G,H}^Σ$ (Böhm polyhedron) introduced by C.Böhm. There is a natural continuous map of $C$ onto the flag …

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $Φ$ is given by a power $β\geq 1$ of a monotone symmetric and homogeneous of degree one function $F$ of the principal curvatures. Under the assumption that $F$ …

The paper solves a specific Dirichlet problem for constant mean curvature surfaces in a particular manifold.

The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended u…

In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these pro…

The paper analyzes oversmoothing in GNNs and quantifies the effects of mixing and denoising.

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,τ)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,τ)$ is such a graph. More specifically: we introduce the definition of a generalized slab in $\mathbb{E…

Polynomial-time test for detecting dense subgraphs in heterogeneous networks.

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admi…

New algorithm tackles multi-agent bandits with heavy-tailed data.

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solu…

The study examines the independence of GKM manifolds and symmetric spaces.

Graph attention networks improve performance on heterogeneous graphs.

New method improves graph neural networks by considering different types of relations in sampling.

Study on flat singularities of area-minimizing currents in codimension one.

Graph neural networks (GNN) has been successfully applied to operate on the graph-structured data. Given a specific scenario, rich human expertise and tremendous laborious trials are usually required to identify a suitable GNN architecture. It is because the performance of a GNN architecture is significantly affected b…

New findings on QHD smoothing for graphs with 3 or 4 large nodes.

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …