This thesis explores GNNs, categorizing them into local and global approaches.
problem Understanding the convergence of global GNNs and connecting local and global approaches.
method Categorization of GNNs into local and global, study of Invariant Graph Networks, connecting local and global approaches, and using local MPNN for graph coarsening.
result Established a connection between local and global GNN approaches.
Non-local GNNs improve performance on disassortative graphs.
problem Efficiency and performance issues in local aggregation for disassortative graphs.
method Proposes a non-local aggregation framework with attention-guided sorting.
result Significantly outperforms previous methods on disassortative graphs.
The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.
problem Identifying which mapping class groups have dense conjugacy classes.
method Developed flux homomorphisms and combinatorial criteria for stability.
result A complete classification for self-similar locally finite graphs and a criterion for stability.
Develops graph uncertainty principles for signals, improving signal reconstruction and analysis.
problem Limits on the concentration of graph signals under dictionary transforms.
method Generalizes Lieb's methods to graph signals, incorporating local structure.
result Local uncertainty principles improve random sampling of graph signals.
Proposes LSGP for better graph signal representation.
problem Local variations in graph process characteristics.
method Locally stationary graph process (LSGP) model.
result LSGP provides accurate signal representations.
New graph convolution captures local features on non-Euclidean grids.
problem Capturing local features on irregular, coarse non-Euclidean grids.
method Low-rank learnable local filters in graph convolutions.
result Proves more expressive than previous spectral graph convolution methods.
Proposes methods for local clustering in attributed graphs.
problem Finding a single cluster concentrated on a specific region in a graph.
method Introduces Graph Unimodality (GU) and Attribute Unimodality (AU) measures, and LOCLU algorithm to optimize Compactness score.
result Local cluster detected by LOCLU concentrates on the region of interest and exhibits unimodal data distribution.
Graphs and local systems count multiwebs.
problem Counting multiwebs in graphs with local systems.
method Using Kasteleyn matrices and web-traces.
result Determinant of Kasteleyn matrix counts multiwebs.
We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains λ-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.
The paper investigates why GNNs struggle to generalize from small to large graphs.
problem Challenges in graph neural networks' ability to generalize across different graph sizes.
method Identified and studied the effect of local structure on size generalization; proposed a novel SSL task.
result GNNs can converge to non-generalizing solutions when there is a discrepancy in local structure.
Graph products inherit Morse local-to-global property from their components.
problem Generalizing local-to-global property to graph products of infinite groups.
method Generalizing maximization procedure for relatively hierarchically hyperbolic groups and showing stable embeddings.
result Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.
Local graph clustering improves with noisy labels, enhancing accuracy and performance.
problem Local graph clustering with noisy labels for node information.
method Constructing a weighted graph with noisy labels and using diffusion-based clustering.
result Diffusion in the weighted graph yields more accurate recovery of target clusters.
Proposes a method to adapt labels on graphs with few labeled nodes.
problem Domain adaptation for graphs with limited labeled nodes.
method Optimization problem solving label transfer using spectral graph wavelets.
result Method yields satisfactory classification accuracy compared to existing methods.
The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.
problem Analyzing heat flow and constants on graphs.
method Introducing concepts, recalling graph theory, and proposing new discrete Morse flows.
result Weak discrete Morse flows for heat flow on finite graphs under suitable assumptions.
Spatial graph representation improves GNN performance.
problem GNNs struggle with distinguishing similar local structures in different graph locations.
method Proposes a spatial graph representation method to distinguish local structures and simplify graph downsampling.
result Proposed graph pooling method achieves competitive results.
Localized signal representation on graph bundles using Fourier analysis.
problem Representing signals on graph bundles with twists.
method Partition of unity and product factorization over the base graph.
result Lifted bases for signal spaces of graph bundle components.
New methods identify local clusters in graphs with few labels.
problem Identifying specific substructures in large graphs without additional structural information.
method Random sampling, diffusion, and overlap analysis of local clusters.
result Proves the correctness of the proposed methods and achieves state-of-the-art results.
Graph ConvNet improves semi-supervised learning on graphs.
problem Efficiently learning from labeled and unlabeled graph data.
method Graph Convolutional Neural Network (GCN) with localized approximation.
result GCN outperforms related methods on citation networks and knowledge graphs.
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
It is shown that for any locally knotted edge of a 3-connected graph in S3, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S3.
New research limits what GNNs can compute and generalizes their performance.
problem Limits of GNNs in computing graph properties and generalization bounds.
method Novel graph-theoretic formalism and data-dependent generalization bounds.
result Proves GNNs can't compute certain graph properties and provides tighter generalization bounds.
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.
CNNs adapted for graph data with fast localized spectral filters.
problem Generalizing CNNs to irregular domains like graphs.
method Spectral graph theory for efficient localized convolutional filters.
result Efficient deep learning system for graph data with linear complexity.
Study p-parabolicity on graphs using various energy functionals.
problem Characterize p-parabolicity on infinite locally summable graphs. method Analyze p-energy functionals and use approximation by finite graphs. result Prove various characterizations of p-parabolicity. Quantum topology methods applied to nonplanar graphs via virtual graphs.
problem Applying quantum topology to nonplanar graphs.
method Defining virtual graphs, extending flow polynomial, and introducing S-polynomial. result Sufficient condition for non-classicality of virtual spatial graphs.
Graphs from knot types help identify unique knots.
problem Identifying knots uniquely.
method Created Reidemeister graphs from knot types and analyzed their properties.
result Graph isomorphism type is a complete knot invariant.
Proposes robust local scaling using conditional quantiles of graph similarities.
problem Spectral analysis sensitivity to parameters and noise.
method Auto-encoding neural network for inferring conditional quantiles of similarity functions.
result Proposed approach outperforms existing methods in spectral clustering and single-example label propagation.
Estimates manifold dimension using local graph structure.
problem Estimating the intrinsic dimension of manifolds from data.
method Regression on local PCA coordinates, focusing on local graph structure.
result Proposed QE and TLS estimators outperform existing methods.
Sharp bounds on diameter and eigenvalues for amply regular graphs.
problem Finding bounds for amply regular graphs' diameter and eigenvalues.
method New ideas relating discrete Ricci curvature to local matching properties, including a novel construction of a regular bipartite graph.
result Sharp diameter and eigenvalue bounds for amply regular graphs.
New relations for vertex polynomial in graphs of any degree.
problem Understanding vertex polynomial in graphs of varying degrees.
method Proved local relations for digons, triangles, quadrilaterals, and pentagons.
result Established new relations for vertex polynomial in graphs of arbitrary degree.
Study of Gordian graphs' behavior at infinity for various local moves.
problem Behavior of Gordian graphs at infinity for different local moves.
method Analysis of unbounded connected components of complements of bounded subsets and finite subsets.
result Complete description of Gordian graphs' behavior at infinity for three families of local moves.
Proposes a graph dynamics prior for more accurate relational inference.
problem Identifying interactions in dynamical systems from observed dynamics.
method Graph Dynamics Prior (GDP) that uses error amplification in non-local polynomial filters.
result Reconstructs graphs more accurately than previous methods, robust to under-sampling.
The study proves sampling-based GNNs can approximate training on full graphs with small subgraphs.
problem Training Graph Neural Networks (GNNs) on large graphs is computationally expensive.
method Theoretical framework using graph local limits to prove approximation of GNN training on small samples.
result Parameters learned from sampling-based GNNs on small subgraphs are close to those on full graphs.
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow on finite graphs.
A framework for federated graph classification over non-IID graphs.
problem Training graph mining models collaboratively across different domains with non-IID graphs.
method Graph Clusters Federated Learning (GCFL) framework, dynamically finding clusters based on GNN gradients, and a gradient sequence-based clustering mechanism (GCFL+).
result Demonstrated effectiveness of GCFL+ in reducing structure and feature heterogeneity among graphs.
Median activation functions improve GNNs by capturing local graph signal behavior.
problem Lack of local nonlinear graph signal encoding in GNNs.
method Proposed median activation functions with support on graph neighborhoods.
result Median activation functions improve GNN capacity with minimal complexity increase.
In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …
New graphs found in CAT(0) group boundaries.
problem Visual boundaries of CAT(0) complexes not homeomorphic.
method Constructed two homeomorphic locally CAT(0) complexes with distinct visual boundaries.
result Visual boundary contains nonplanar graph in one case, not in the other.
Graph-CNN for 3D point cloud classification tackles non-regular graph topology.
problem Classifying 3D point cloud data with non-regular graph topology.
method Developed PointGCN combining localized graph convolutions and graph downsampling.
result Achieves competitive performance on 3D object classification benchmark ModelNet.
GNNs may be limited by graph topology, affecting their learning outcomes.
problem Understanding how graph topology influences GNN behavior and performance.
method Investigating the interaction between local topological features and GNN message-passing schemes.
result Locally similar neighborhoods can lead to consistent node representations, affecting GNN performance.
Graph neural network framework learns graph representations from node features and local structures.
problem Lack of hierarchical pooling to preserve graph structure in graph neural networks.
method Introduces a pooling operator based on graph Fourier transform to combine node features and local structures.
result Framework $\m$ improves graph classification performance on 6 benchmarks.
Paper presents a novel approach for global feature aggregation in Graph Neural Networks.
problem Graphs lack a straightforward way to perform non-local feature aggregation like images and texts.
method Utilizes Latent Fixed Data Structure (LFDS) to aggregate feature vectors from local extraction.
result Proposed methods achieve competitive or better results with linear computational complexity.
This work investigates how GCNs should handle local structure discrepancies in testing nodes.
problem GCNs assume homophily but real graphs often have discrepancies in local structure.
method Using causal graph analysis, the study intervenes the graph structure to assess the local structure's impact on predictions.
result The method effectively enhances GCN predictions by eliminating local structure discrepancies.
This paper improves nearest neighbor search by learning optimal routing functions.
problem Local minima issues in greedy routing on similarity graphs.
method Learn routing function that considers global graph structure.
result Significant improvement in search performance via learning.
This paper introduces Haar convolution for GNNs to reduce computational cost.
problem High computational cost in GNNs for large graph sizes.
method Introduces Haar basis for graph convolution and Fast Haar Transforms.
result State-of-the-art results on graph-based regression and node classification tasks.
The paper characterizes chordal graphs via edge deletions and finds a local minimum spanning tree algorithm.
problem Characterizing chordal graphs and finding efficient minimum spanning trees.
method Focus on exposed edges, characterize chordal graphs via deletions, and use local properties to modify Kruskal's algorithm.
result A modified Kruskal's algorithm for weighted chordal graphs is local and efficient.
In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…
Method solves Gaussian graphical models on ladder graphs efficiently.
problem Solving Gaussian graphical models on ladder graphs efficiently.
method Proposes a method that depends on the position of zeros in local covariance matrices.
result Efficiently solves Gaussian graphical models on ladder graphs under certain conditions.