Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.
problem Understanding automorphisms and subdivisions of Helly graphs.
method Simple fine simplicial subdivisions and explicit simplicial models of the injective hull.
result Any automorphism of a Helly graph is either elliptic or hyperbolic, with rational translation lengths.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bico…
The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.
problem Actions by automorphisms of finitely generated groups on nonpositively curved complexes without fixed points.
method Use of Helly graphs and geodesic clique paths to prove ellipticity results.
result Finitely generated torsion groups cannot act without fixed points on nonpositively curved spaces.
The article explores metrics on buildings and symmetric spaces, proving injectivity and proper actions.
problem Injectivity of metrics on buildings and symmetric spaces.
method Analyzing norms, Helly property, and group actions.
result Most classical buildings and symmetric spaces can be endowed with injective metrics.
The study proves fixed-point theorems for groups acting on CAT(0) spaces.
problem Finding fixed points for groups acting on CAT(0) spaces.
method Bootstrapping technique with Helly-type theorems to prove intersections of fixed-point sets.
result Lower bounds on the smallest dimension for groups to act on CAT(0) spaces without global fixed points.
Paper characterizes classes for which optimal sample complexity can be achieved by proper learning algorithms.
problem Characterizing classes for which optimal sample complexity can be achieved by proper learning algorithms.
method Identifying dual Helly number and its relation to proper learning algorithms.
result Optimal sample complexity can be achieved by proper learning algorithms for classes with bounded dual Helly number.
Theory of space-time currents for geometric evolutions.
problem Analysis of geometric evolutions driven by dislocations.
method Development of space-time integral currents with bounded variation, introduction of Lipschitz deformation distance.
result Agreement of Lipschitz deformation distance with integral Whitney flat metric for boundaryless currents.
This paper shows that a construction, which was introduced by Piotr Minc in connection with a problem that came from Helly type theorems and that allows to replace three PL-arcs with a "sheltered middle path", can in the case of general (non-PL) paths result in the topologist's sine curve.
Unified solution to Goodman-Pollack transversal problem using matroids and topology.
problem Existence of an affine k-dimensional transversal to convex sets.
method Matroidal joins and topological methods.
result Unified solution including colorful Helly theorem and Holmsen's theorem.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
Reduces conjecture for Artin groups to simpler cases.
problem Proving K(π,1) for Artin groups with specific spherical parabolics. method Reduces to simpler cases, uses injective metric spaces, combinatorial convexity, and Bestvina-type inequalities.
result Deduces K(π,1) conjecture for specific Artin groups. New group not biautomatic, geometrically constructed.
problem Constructing a non-biautomatic group.
method Using a CAT(0) hierarchically hyperbolic group and geodesic currents.
result First example of a non-biautomatic CAT(0) group of flat-rank 2.
Study of f-neighbors in Riemannian manifolds, proving infinite set of distances.
problem Exploring variations of Hopf theorem in Riemannian manifolds.
method Investigates continuous maps of compact Riemannian manifolds to Rm and introduces f-neighbors. result Set of distances realized as visual f-neighbors is infinite. We improve the bound on Kühnel's problem to determine the smallest n such that the k-skeleton of an n-simplex Δn(k) does not embed into a compact PL 2k-manifold M by showing that if Δn(k) embeds into M, then n≤(2k+1)+(k+1)βk(M;Z2). As a consequence we obtain improved Radon and He…
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.
Proposes MGMN for end-to-end graph similarity learning.
problem Lack of cross-level interactions in graph similarity learning.
method Multi-level graph matching network (MGMN) combining node-graph matching and siamese graph neural networks.
result MGMN outperforms state-of-the-art models on graph-graph classification and regression tasks.
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
MxPool learns graph features from diverse graphs using a hierarchical structure.
problem Learning graph features from diverse graphs with varying properties and sizes.
method MxPool uses a multiplex structure with multiple graph convolution/pooling networks in a hierarchical learning structure.
result MxPool outperforms state-of-the-art methods on graph classification benchmarks.
Study the geometry of graph product extension graphs.
problem Properties of graph products.
method Introduce and study the extension graph of graph products of groups.
result Extension graph is isomorphic to crossing graph of a quasi-median graph and exhibits asymptotic dimension similar to quasi-trees.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
Customized-GNN generates model-specific for each graph.
problem Graphs in the same dataset have distinct structures.
method Proposes Customized-GNN framework to generate model-specific for each graph.
result Demonstrates effectiveness on various graph classification benchmarks.
Graph embedding leaks sensitive graph properties and subgraphs.
problem Privacy risks in graph embedding sharing.
method Three inference attacks and a defense mechanism.
result High accuracy in inferring graph properties and subgraphs.
Characterizes graphs with leveled embeddings and introduces new graph invariants.
problem Understanding the properties of leveled embeddings in spatial graphs.
method Characterization of graphs with leveled embeddings, introduction of new invariants.
result Characterization of graphs with low level number and determination of specific invariants for complete graphs and complete bipartite graphs.
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
Two new methods improve graph embedding without needing a complete graph structure.
problem Graph autoencoders' performance depends on the adjacency matrix quality.
method BAGE and VBAGE: unsupervised graph embedding via adaptive graph learning.
result The methods expand GAEs' applicability to datasets without graph structure.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
Develops method to create non-Abelian Ricci-flat graphs via bundles.
problem Creating non-Abelian Ricci-flat graphs.
method Develops systematic way via graph bundles with constraints.
result Non-trivial graph bundles are not isomorphic to product of base and fiber.
New method uses graph generative models for graph classification.
problem Graph classification for non-relational i.i.d. data.
method Derive classification formulas from GGM, train generative graph auto-encoder model.
result New conditional ELBO for training graph auto-encoder model.
MathNet uses wavelets for graph representation and learning.
problem Graph Neural Networks (GNNs) for graph classification and regression.
method Multiresolution Haar-like wavelets, graph convolution, and pooling.
result MathNet achieves notable accuracy gains on graph classification and regression tasks.
Unified framework for graph coarsening using node features and graph matrices.
problem Dimensionality reduction of large graphs while preserving node features.
method Optimization-based framework that unifies graph learning and dimensionality reduction.
result The learned coarsened graph is ε-similar to the original graph, where ε is a small positive number.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.
Study classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.
PSimGNN partitions graphs into subgraphs for efficient graph similarity computation.
problem Efficiently compute graph similarity scores for large graphs.
method Graph partitioning followed by subgraph-level and node-level comparisons using a graph neural network.
result PSimGNN outperforms state-of-the-art methods in graph similarity computation tasks.
We present graph wavelet neural network (GWNN), a novel graph convolutional neural network (CNN), leveraging graph wavelet transform to address the shortcomings of previous spectral graph CNN methods that depend on graph Fourier transform. Different from graph Fourier transform, graph wavelet transform can be obtained …
This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
Graph Convolutional Neural Networks (Graph CNNs) are generalizations of classical CNNs to handle graph data such as molecular data, point could and social networks. Current filters in graph CNNs are built for fixed and shared graph structure. However, for most real data, the graph structures varies in both size and con…
We introduce a novel approach to graph-level representation learning, which is to embed an entire graph into a vector space where the embeddings of two graphs preserve their graph-graph proximity. Our approach, UGRAPHEMB, is a general framework that provides a novel means to performing graph-level embedding in a comple…
Graph Cascades rewire graphs to improve structure-aware learning.
problem Improving graph neural networks and transformers for structure-aware learning.
method Graph Cascades uses contagion-based diffusion processes to construct an auxiliary graph with reinforced edges.
result Graph Cascades improves node-classification benchmarks across various graph types.
CTGCN learns dynamic graph embeddings preserving both local and global graph structure.
problem Learning node representations for evolving graphs while preserving both local and global graph structure.
method CTGCN uses k-core based temporal graph convolutional network to learn dynamic graph embeddings.
result CTGCN outperforms existing methods in link prediction and structural role classification.
The dominant graph neural networks (GNNs) over-rely on the graph links, several serious performance problems with which have been witnessed already, e.g., suspended animation problem and over-smoothing problem. What's more, the inherently inter-connected nature precludes parallelization within the graph, which becomes …
Graphs can be fooled by small edge changes, but this work protects them.
problem Adversaries can manipulate graph data to mislead graph classification models.
method We introduce a smoothed graph classification model with a robustness guarantee.
result The smoothed model maintains consistent predictions under small adversarial perturbations.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
SpeqNets improve graph neural networks by scaling and adapting to graph sparsity.
problem Graph neural networks struggle with permutation-equivariant functions and scalability to large graphs.
method Introducing sparsity-aware, permutation-equivariant graph networks with heuristics for graph isomorphism.
result Significantly improved predictive performance and reduced computation times compared to existing methods.
Paper converts graph learning to lifelong learning.
problem Learning graphs in a streaming fashion.
method Feature graph topology, converting node classification to graph classification.
result FGN achieves superior performance in lifelong human action recognition and feature matching.
Graph embedding is an effective method to represent graph data in a low dimensional space for graph analytics. Most existing embedding algorithms typically focus on preserving the topological structure or minimizing the reconstruction errors of graph data, but they have mostly ignored the data distribution of the laten…