The paper analyzes the Spectral Method for clustering data points on Union of Subspaces.
problem Clustering data points on Union of Subspaces.
method Constructing a Random Geometry Graph (Subspace Clustering) and analyzing it using spectral methods.
result Established a theory to analyze the Spectral Method's efficiency on Union of Subspaces.
Study on connectivity and geometry of random Coxeter groups.
problem Connectivity threshold for square percolation on random graphs.
method Probabilistic combinatorics and techniques from geometric group theory.
result Determines connectivity threshold and cubical coarse median structure for random Coxeter groups.
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 . Survey on strong convergence in random matrices and its applications.
problem Understanding convergence of random matrices to operators.
method Analysis of operator norms of noncommutative polynomials.
result New insights and applications in random graphs, geometry, and operator algebras.
The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.
problem Uniformization and boundary theory of random infinite ideal hyperbolic polyhedra and their dual graphs.
method Combinatorics, geometry, analysis, and random walks perspectives.
result Characterization of the ICP type of IAG and convergence of simple random walk to the boundary.
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…
Study finds significant instability in node embeddings due to randomness.
problem Stability of node embeddings under random variations.
method Evaluated five node embedding algorithms (HOPE, LINE, node2vec, SDNE, GraphSAGE) on synthetic and empirical graphs.
result Significant instability in embedding spaces and downstream task accuracy.
Study on detecting and recovering hidden dense cycles in random graphs.
problem Detecting and recovering hidden dense cycles in random graphs.
method Information-theoretic analysis of thresholds for detection and recovery.
result Characterization of information-theoretic thresholds for detection and recovery.
Convex optimization method infers latent structure in random dot product graphs.
problem Inferring latent probability matrix of random dot product graphs.
method Conic programming with nuclear norm regularization.
result Asymptotic consistency of probability estimates and recovery of latent structure.
Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.
problem Understanding geometric properties of graphs with non-negative Ollivier-Ricci curvature.
method Analyzing the geometric properties of graphs with non-negative Ollivier-Ricci curvature, proving subexponential growth and diffusive random walk.
result For graphs with bounded degrees and non-negative Ollivier-Ricci curvature, the average log-volume growth and random walk displacement are subexponential.
The paper predicts responses on out-of-sample nodes using latent positions on unknown curves.
problem Predicting responses on out-of-sample nodes with latent positions on unknown curves.
method Manifold learning and graph embedding technique using latent positions.
result Convergence guarantees for predicting responses on out-of-sample nodes.
New methods learn from single graphs, improving transductive node classification.
problem Statistical foundations of transductive learning for single graphs.
method Developed new concentration-of-measure tools for large graphs.
result Achieved optimal nonparametric rate of N − 1 / 2 N^{-1/2} N − 1/2 for single graph learning. In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…
This paper tackles matching two complete graphs with correlated edge weights in geometric models.
problem Matching two complete graphs with edge weights correlated through latent geometries.
method Derives an approximate maximum likelihood estimator for recovering hidden vertex correspondence.
result The estimator provably achieves perfect recovery under certain noise conditions.
Bayesian models for networks are often misspecified, leading to overconfident inference.
problem Real-world networks violate assumptions of geometry and link function in latent space models.
method Proposes a generalized posterior framework for random geometric graphs, using Link-Sequential R-SafeBayes to adaptively tune posterior regularization.
result Improved calibration and better link prediction performance demonstrated on synthetic and real-world networks.
A new graph-based clustering method for moderate-dimensional data.
problem Performance degradation of existing graph-based clustering methods in high dimensions.
method Introduces UN-CCDs using NND-based MC-SRT for covering radii determination.
result UN-CCDs provide stable and competitive performance in moderate-sized datasets.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
A parametrization of hypergraphs based on the geometry of points in R d \mathbf{R}^d R d is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or M…
FGN models networks with fractal structures using Gaussian Multiplicative Chaos.
problem Modeling networks with fractal structures.
method FGN model based on Gaussian Multiplicative Chaos.
result FGNs reveal distinct scaling patterns in edge and clique counts.
Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
problem Uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
method Analysis of random walks on geometric and directed kNN graphs, using concentration tools and differential geometry.
result Uniform convergence of k k k NN Laplacians to diffusion Laplacian, without continuity of transition kernel. Constructs manifolds from quantum codes with novel geometric properties.
problem Creating manifolds with specific geometric constraints.
method Reverse engineering manifolds from quantum code chain complexes.
result First examples of power law Z 2 \mathbb{Z}_2 Z 2 systolic freedom. The paper introduces heterogeneous manifolds for better graph embeddings.
problem Graph embeddings in Euclidean spaces often fail to capture the curvature of real-world graphs.
method The authors propose heterogeneous rotationally-symmetric manifolds with a radial dimension to account for varying curvature.
result The method improves graph embeddings by better preserving high-order structures and heterogeneous random graphs.
Riemannian metric matching learns the geometry of high-dimensional datasets using neural networks.
problem Estimating the geometry of high-dimensional datasets from samples
method Riemannian metric matching using neural networks
result Riemannian metric matching rivals or improves k k k -NN-based diffusion geometry estimators New framework for neural networks converging to low loss without overparameterization.
problem Training deep neural networks without overparameterization assumptions.
method Construction of random sparse lifts and analysis using algebraic topology and random graph theory.
result Provable convergence to low loss for large sparse neural networks.
New methods for calculating curvature in graph theory.
problem Calculating curvature in graphs and random walks.
method Analyzing continuous and discrete-time Ollivier-Ricci curvatures of weighted graphs.
result Generalized existence and properties of Ollivier-Ricci curvature for various random walks.
We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ > 0 δ> 0 δ > 0 if p ≥ ( 1 / 2 + δ ) log n n , p \ge \frac{(1/2 + δ) \log n}{n}, p ≥ n ( 1/2 + δ ) l o g n , then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1 1 1 . We est…
The study explores discrete versions of Riemannian geometry structures on manifolds.
problem Understanding the relationship between discrete structures and continuous Riemannian geometry.
method Surveying and analyzing discrete counterparts of Riemannian geometry concepts on graphs and simplicial complexes.
result Recent developments include Cheeger type inequalities for higher-dimensional simplicial complexes and Floer type constructions.
New method beats volumetric barrier for manifold recovery.
problem Reconstructing latent geometry from noisy distances.
method Orthogonal Ring Distance Estimation Routine (ORDER).
result Achieves pointwise distance estimation of order n − 2 / ( d + 5 ) n^{-2/(d+5)} n − 2/ ( d + 5 ) . New method for faster graph parameter inference from large random Kronecker graphs.
problem Efficiently infer graph parameters from large random Kronecker graphs.
method Decompose adjacency matrix into signal and noise components, then use denoising and solving approach.
result Proposed method achieves comparable or better performance than existing methods at lower computational cost.
GraphMoE generates random graphs using neural networks and graphlets.
problem Learning generative models for random graphs.
method GraphMoE uses a neural network trained with graphlets and subgraph counts to match the distribution of random graphs.
result GraphMoE can generate graphs that mimic various real-world datasets and fool graph classifiers.
The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starti…
A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, t…
Graph Neural Networks struggle on random graphs without node identifiers.
problem Graph Neural Networks' limitations on random graphs without node identifiers.
method Study of Graph Neural Networks and Structural Graph Neural Networks convergence on large random graphs.
result Structural Graph Neural Networks are more powerful and universal than Graph Neural Networks on random graphs.
We develop a computationally efficient method to estimate Ollivier-Ricci curvature.
problem Computational infeasibility of evaluating Ollivier-Ricci curvature on large graphs.
method Derive explicit transfer moduli between OR and BF curvatures, construct lazy transport envelopes, and use cross-edge matching.
result Deterministic bounds for OR curvature parameterized by local graph combinatorics, reducing complexity to worst-case O(max_v deg(v)^1.5).
In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erdös-Renyi random graphs, geometric random graphs, expanders and random k k k -regular graphs. Moreover we show that …
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.
CuBAS selects informative data points based on curvature for better classification.
problem Lack of efficient sampling strategies for maximizing dataset informativeness.
method Information-geometric framework using curvature scores to select labeled data.
result Consistent and statistically significant improvements over random and uncertainty-based sampling.
Researchers calculate spectral dimension of complex networks using renormalization group theory.
problem Understanding diffusion properties in complex systems.
method Renormalization group theory applied to graph Laplacians of simplicial complexes.
result Spectral dimension decreases with randomness in topological structure.
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.
Unified curvature for hypergraphs from Ollivier-Ricci.
problem Generalizing curvature to hypergraphs.
method Developed ORCHID framework to generalize Ollivier-Ricci curvature to hypergraphs.
result ORCHID curvatures have favorable theoretical properties and are scalable for hypergraph tasks.
Sharp threshold found for Frechet mean of inhomogeneous graphs.
problem Finding the Frechet mean of inhomogeneous Erdos-Renyi random graphs.
method Thresholding the expected adjacency matrix of the ensemble.
result The Frechet mean graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold.
New graph kernel scales well with graph size and number, achieving state-of-the-art performance.
problem Graph kernels lose structure information when representing graphs.
method Proposes a positive-definite global alignment graph kernel using random features and random graph embeddings.
result Achieves quasi-linear scalability with respect to graph size and number.
Topology helps estimate chromatic numbers of random graphs on spheres.
problem Estimating chromatic numbers of random graphs on spheres.
method Topology, specifically connectivity of Lóvasz's neighborhood complex.
result Connectivity bound is useful in dimensions 1 and 2, but generally poor.
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.
Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n goes to infinity. This includes random ri…
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.
A new method improves graph random features with quasi-Monte Carlo techniques.
problem Improving the accuracy of graph random features.
method Induces negative correlations in random walks using antithetic termination.
result Strong theoretical guarantees on lower-variance estimators of the Laplacian kernel.
This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs. VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the n…