Study of graphs interpolating curve and pants graphs, providing formulae and geometry classifications.
problem Understanding the large-scale geometry of graphs connecting curve and pants graphs.
method Developed explicit formulae for quasi-flat ranks and classified geometries using twist-free graphs of multicurves.
result Explicit formulae for quasi-flat ranks and classification of geometries into hyperbolic, relatively hyperbolic, and thick cases.
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.
Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
problem Understanding the geometry of graph dynamical systems with odd interactions.
method Proved geometry and stability of manifolds governed by graph homology and coverings.
result Derived upper and lower bounds on the dimension of the equilibrium set.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
In a graph convolutional network, we assume that the graph G G G is generated wrt some observation noise. During learning, we make small random perturbations Δ G ΔG Δ G of the graph and try to improve generalization. Based on quantum information geometry, Δ G ΔG Δ G can be characterized by the eigendecomposition of the graph Laplaci…
Liouville theorems extended to graphs with bounded geometry.
problem Ancient solutions of subexponential growth on graphs.
method Extended Mosconi's results to graphs with bounded geometry.
result Nonnegative ancient solutions are stationary and harmonic.
Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
Spectral graph sparsification preserves geometry of GNN embeddings.
problem Maintaining geometric properties of graph neural network embeddings during sparsification.
method Proving spectral sparsification preserves squared pairwise distances, class means, and covariance structure in embedding space.
result Spectral sparsification preserves the geometry of learned embeddings in GNNs.
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Enhances knowledge graph completion with mixed geometry tensor factorization.
problem Capturing nuanced distributional properties in knowledge graphs.
method Combines Euclidean and hyperbolic geometries for tensor factorization.
result Improves link prediction accuracy with fewer parameters.
Study of graphs from hexagon decompositions of surfaces.
problem Understanding geometric properties of hexagon decompositions.
method Define and analyze graphs associated with hexagon decompositions of surfaces.
result Quasi-isometric relationships between studied graphs and known groups.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
Study shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
Graph Ricci flow reveals hidden hierarchies in stock market correlations.
problem Detecting hidden structures in the complex stock market graph.
method Using graph Ricci curvature and flow techniques to analyze the NASDAQ 100 index.
result Algorithm detects hidden hierarchies, community behavior, and clustering in financial markets.
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …
Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.
problem Reconstructing smooth real algebraic maps onto curves with specific Reeb graphs.
method Developed a method to reconstruct functions from general finite graphs, focusing on curves.
result Reconstructed functions from prescribed Reeb graphs, providing a new approach in real algebraic geometry.
Study the geometry of nonseparating curves on surfaces with punctures.
problem Investigate the geometry of nonseparating curves on surfaces with punctures.
method Study finite covers and lifts of nonseparating curves, analyze bicorn curves and laminations.
result Lifts of nonseparating curves span quasiconvex subgraphs in the nonseparating curve graph of covers.
HGCN uses hyperbolic geometry to improve graph node embeddings.
problem Distortion in Euclidean embeddings of real-world graphs.
method Derives GCN operations in hyperbolic space and maps Euclidean features to hyperbolic embeddings.
result HGCN achieves up to 63.1% error reduction in ROC AUC for link prediction.
Study G 2 G_2 G 2 -manifolds from symplectic S U ( 3 ) SU(3) S U ( 3 ) -manifolds with T 2 T^2 T 2 -symmetry.
problem Understanding G 2 G_2 G 2 -manifolds from symplectic S U ( 3 ) SU(3) S U ( 3 ) -manifolds. method Cohomological lifting of multi-toric graphs.
result Compact part of G 2 G_2 G 2 -moment graph can be obtained cohomologically from the base. A new algorithm improves sampling for graph learning models.
problem Euclidean proposals struggle near the boundary of PSD matrices.
method ConeMALA, a geometry-aware Langevin algorithm.
result ConeMALA achieves higher ESS/sec and stable diagnostics.
Neural networks' feature geometry evolves like discrete Ricci flow.
problem Understanding neural feature representations and their geometric transformations.
method Approximating input manifold with geometric graphs and analyzing their evolution during training.
result Neural feature geometry evolves like discrete Ricci flow, with nonlinear activations playing a crucial role.
The study combines graph-minors and metric spaces, answering some questions and conjectures.
problem Whether geodesic metric spaces without a fat H H H minor are quasi-isometric to graphs without H H H minor. method Combining graph-minors and coarse geometry, answering affirmatively for small H H H . result Affirmative answer for small H H H in the problem statement. We prove that the Cayley graph and the coset geometry of the von Dyck group D ( a , b , c ) D(a,b,c) D ( a , b , c ) are linked by a vertex-to-edge duality.
Hyperbolic GNNs improve graph data learning.
problem Learning from graph-structured data.
method Proposes a novel GNN architecture for Riemannian manifolds.
result Hyperbolic GNNs lead to substantial improvements on benchmark datasets.
Study large-scale geometry of graph braid groups via cubical structures.
problem Classify and understand the quasi-isometry of graph braid groups.
method Exploit cubical structures to relate hyperbolicity, undistorted subgroups, and group decompositions.
result Complete classification of graph braid groups quasi-isometric to free groups.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
This is a survey paper. We study the Ricci curvature and spectrum of graphs, as well as the exterior forms and deRahm cohomology on graphs.
New method separates graph structure from node attributes to recover lost signal.
problem Standard representation learning on attributed graphs merges incompatible metric spaces, leading to geometrically flawed alignment.
method Custom variational autoencoder that separates manifold learning from structural alignment.
result Transforms geometric conflict into interpretable structural descriptor, uncovering connectivity patterns and anomalies.
Graph kernels for metric graphs using tropical algebra.
problem Comparing graphs representing different metric spaces.
method Purely based on geometry and topology, invariant under edge subdivision.
result Capture complementary geometric and topological information.
The study compares lamplighter graphs up to quasi-isometry using coarse topology.
problem When do two lamplighter graphs have the same coarse geometry?
method Inspired by topology, the approach involves techniques to compare lamplighter graphs up to quasi-isometry.
result Efficient comparison methods for lamplighter graphs up to quasi-isometry.
The paper solves graph realization problems for Reeb graphs of Morse functions.
problem Realizing graphs as Reeb graphs with specific preimage configurations.
method Constructing Morse functions with prescribed preimages.
result Solved realization problems for certain types of graphs.
We show that there are minimal graphs in R^{n+1} whose intersection with the portion of the horizontal hyperplane contained in the unit ball has any prescribed geometry, up to a small deformation. The proof hinges on the construction of minimal graphs that are almost flat but have small oscillations whose geometry we c…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the construction is sufficiently regular and uniform, then the notions of metric, geodesic and…
We explore the use of graph networks to deal with irregular-geometry detectors in the context of particle reconstruction. Thanks to their representation-learning capabilities, graph networks can exploit the full detector granularity, while natively managing the event sparsity and arbitrarily complex detector geometries…
Metric graphs have subgraphs with entropy at least λ.
problem Finding subgraphs with high entropy in metric graphs.
method Proving existence of subgraphs with entropy at least λ for graphs of rank r with entropy 1.
result Metric graphs have subgraphs with entropy at least λ.
Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on this theory, we demonstrate the efficiency of Subspace Clustering in fairly broad…
GS-B 3 ^3 3 SE improves label shift estimation by smoothing priors on a graph.
problem Label shift adaptation when source and target distributions share conditional but not marginal probabilities.
method Graph-Smoothed Bayesian Black-Box Shift Estimator (GS-B 3 ^3 3 SE) places Laplacian-Gaussian priors on log-priors and confusion-matrix columns tied by a label-similarity graph. result GS-B 3 ^3 3 SE produces a tractable posterior with HMC or Newton-CG schemes, proving identifiability, contraction, and robustness. This paper connects graph curvature to community structure.
problem Understanding the relationship between network curvature and community formation.
method Defining curvature on networks and analyzing its relation to community structure.
result Apriori bounds on the curvature of intercommunity edges.
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
Enhances graph modeling with hyperbolic geometry and variational inference.
problem Challenges in modeling relational data with complex dependencies.
method Semi-implicit hierarchical variational Bayes with Poincaré embedding and mutual information regularization.
result Improves graph representation quality and flexibility in edge prediction and node classification.
Neural network learns from higher-order connections in molecules.
problem Graph neural networks fail to account for local and hidden structures in graphs.
method Developed a neural network that can pass messages and aggregate information across higher-order paths.
result The model improves molecular property prediction.
We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set the bandwidth by optimizing the Laplacian's ability to preser…
The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang find a sufficient condition for …
Study reveals limits of detecting local geometry in random graphs.
problem Detecting local geometry in random graphs with hidden communities.
method Introduced model and used information-theoretic and computational limits to investigate detection.
result Detection threshold determined at d = Θ ~ ( k 2 ∨ k 6 / n 3 ) d = \widetildeΘ(k^2 \vee k^6/n^3) d = Θ ( k 2 ∨ k 6 / n 3 ) for fixed p p p . A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their a…