GANs can analyze graph topology, ranking edge sets by their importance.
problem Capturing topological features of graphs using GANs.
method Leveraging hierarchical connectivity structure of graphs, GANs rank edge sets by their contribution to topology reconstruction.
result GANs can preserve important topological features in graphs.
Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
New method reduces spatial graphs while preserving their topological features.
problem Finding a smaller spatial graph with the same structure.
method Topological spatial graph coarsening approach based on triangle-aware graph filtration.
result Significant reduction in graph size while preserving topological information.
Characterizes groups for Petersen graph embeddings.
problem Identifying symmetry groups of the Petersen graph.
method Analyzes embeddings of the Petersen graph in S^3.
result Characterizes all possible symmetry groups.
The paper calculates topological complexity for non-tree graphs and banana graphs.
problem Determining topological complexity for non-tree graphs and banana graphs.
method Analyzes fully articulated graphs and banana graphs, completing previous results for trees.
result Shows that unordered configuration spaces can have lower topological complexity than ordered ones.
Persistent homology enhances graph classification by capturing long-range graph properties.
problem Lack of formal assessment of persistent homology in graph learning.
method Brief introduction and theoretical discussion of persistent homology in graph context, followed by empirical analysis.
result Persistent homology improves graph classification, especially for data with prominent topological structures.
Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
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.
Novel TRI-GNN framework improves graph classification robustness.
problem Graph neural networks suffer from over-smoothing and vulnerability to graph perturbations.
method Integrates higher-order graph information via persistent homology and local graph structure learning.
result TRI-GNN outperforms state-of-the-art baselines on node classification tasks.
Abstract: Topological quantum field theory connects graph evaluations to polynomial identities.
problem Graph evaluations in topological quantum field theory.
method Relates SO(3) topological quantum field theory trace evaluations to topological Tutte polynomial evaluations.
result Generalizes the Tutte golden identity for graphs on the torus.
Detects graph topology changes from noisy signals using prior spectral information.
problem Detecting changes in graph topology from graph signals.
method Leverages graph filtering and subspace detection to distill problem into a CUSUM-based algorithm.
result Demonstrates the effectiveness of incorporating prior spectral signatures for change-point detection.
TOGL adds topological info to GNNs, improving graph and node classification.
problem Graph neural networks lack substructure awareness, especially cycles.
method Integrates global topological information using persistent homology.
result Improves predictive performance for graph and node classification.
Study groups that can embed Heawood graph in 3D space.
problem Classifying topological symmetry groups of the Heawood graph.
method Analyzing embeddings of the Heawood graph in S3. result Identified all possible groups as topological symmetry groups.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
Paper tackles dynamic graph topology identification in time-varying graphs.
problem Dynamic graph topology identification in time-varying graphs.
method Proposes an online algorithm for time-varying optimization, with intrinsic temporal regularization.
result Demonstrates performance on Gaussian graphical model problem.
Enhances graph embeddings by preserving graph topology.
problem Node2vec struggles to recreate the topology of input graphs.
method Introduces a topological loss term to Node2vec, aligning the persistence diagram of the embedding to that of the input graph.
result Reconstructs both geometry and topology of input graphs.
TAGCN improves graph CNN performance without approximation.
problem Performance loss in spectral graph convolutional neural networks.
method Topology adaptive graph convolutional network (TAGCN) with adaptive filters.
result TAGCN outperforms existing spectral CNNs on various datasets.
Graph braid groups' complexity stabilizes for most graphs.
problem Stabilization of topological complexity in graph braid groups.
method Geometric lower bounds on configuration spaces.
result Topological complexity stabilizes for most graphs.
Graph neural network using Beltrami flow for feature and topology evolution.
problem Efficient feature learning and topology evolution on graphs.
method Discretized Beltrami flow applied to graph neural networks with positional encodings.
result Achieves state-of-the-art results on various benchmarks.
This paper determines all possible topological symmetry groups of generalized Petersen graphs.
problem Identifying all topological symmetry groups of generalized Petersen graphs.
method Analyzing embeddings of generalized Petersen graphs in S3 and considering homeomorphisms. result All groups that can be topological symmetry groups of generalized Petersen graphs are identified.
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…
Develops methods to learn graphs with specific topology properties.
problem Learning graphs with desired topology properties (e.g., k-partite) is non-convex. method Decomposes problem into GTI and GWE steps; GTI selects feasible graph topology, GWE estimates graph weights.
result Error bound on GWE step as a function of GTI step error, indicating GTI should threshold similarity matrix.
Generative model predicts multiple brain graphs from one, preserving topology.
problem Predicting multiple brain graphs from a single one, preserving topology.
method MultiGraphGAN architecture, graph adversarial auto-encoder, cluster-specific decoders, topological loss.
result Significantly outperformed variants in multi-view brain graph generation.
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.
A new topology improves decentralized learning efficiency and accuracy.
problem Finding efficient decentralized learning topologies with fast consensus and low maximum degree.
method Proposed the Base-(k+1) Graph topology for decentralized learning. result The Base-(k+1) Graph enables faster convergence and better communication efficiency than the exponential graph. Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S3. result All graphs in the Heawood family are intrinsically chiral.
The paper investigates if graph embeddings capture key topological features.
problem Exploring if graph embeddings approximate traditional vertex level graph features.
method Predicting known topological features from graph embeddings using supervised and unsupervised methods.
result Several topological features are approximated by the embedding space, providing insight into how graph embeddings function.
AdaCGP learns dynamic graph topology from time series data, improving over existing methods.
problem Learning dynamic graph topology from time-varying signals, especially in real-time applications.
method AdaCGP is a sparsity-aware adaptive algorithm that recursively estimates the Graph Shift Operator (GSO) through variable splitting.
result AdaCGP outperforms state-of-the-art methods in GSO estimation, achieving improvements exceeding 83%.
This work characterizes topological descriptors of graph products and their expressive power.
problem Capturing multiscale structural information in graph products using topological descriptors.
method Analysis of various filtrations on graph products, including Euler characteristic and persistent homology.
result Persistent homology of graph products contains more information than individual graphs.
Proposes a method to infer complex network topologies from multiple graphs.
problem Learning multiple graph Laplacian matrices from heterogeneous graph signals with intricate topological patterns.
method Structured fusion regularization and ADMM algorithm for efficient computation.
result Establishes a non-asymptotic bound of the estimation error and reflects the effect of key factors on convergence rate.
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in th…
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
The study finds all possible heights for transformation groups on graphs.
problem Determining the possible heights of transformation groups on graphs.
method Proved the existence of a topological graph X for all finite p≥0. result For all finite p≥0, there exists a graph X such that the set of heights is {p,p+1,p+2,…}∪{+∞}. 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.
Constructs hyperbolic manifolds with surfaces of high topological index.
problem Creating hyperbolic manifolds with surfaces of high topological index.
method Using graph theory and topological properties, constructs surfaces in hyperbolic manifolds.
result Proves existence of hyperbolic manifolds containing surfaces of arbitrarily high topological index.
Graph kernel uses Ricci curvature for comparison.
problem Graph comparison without node attributes.
method Edge curvature distribution for graph kernel.
result Graphs can be compared using topology alone.
Survey on spatial graphs with few vertices and edges.
problem Topology of spatial graphs with limited vertices and edges.
method Survey and focus on Brunnian θ-graphs.
result Survey reveals insights into spatial graphs.
Paper presents voxel graph operators for vector data models.
problem Efficient conversion and analysis of geometric models.
method Topological voxelization, graph construction, differential operator derivation.
result Discrete differential and integral operators from voxel complexes.
Research predicts XRP price anomalies using graph topologies.
problem Forecasting extreme price movements in XRP cryptoasset.
method Analyzed topological features of XRP transaction graphs.
result Topological features indicate extreme price surges.
Graph neural networks improve topology control of power grids.
problem Grid congestion due to renewable energy and electrification.
method Investigated the effect of graph representation on GNN effectiveness for topology control.
result Heterogeneous graph representation outperforms homogeneous in topology control tasks.
Self-complementary graphs have complete minors.
problem Finding complete minors in self-complementary graphs.
method Analyzing topological properties of self-complementary graphs.
result Self-complementary graphs contain K⌊2n+1floor minors. Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Graph Convolutional Networks improved with topological features for better accuracy.
problem Improving Graph Convolutional Networks for node classification.
method Using topological features of nodes and adjacency matrices with distant nodes of similar topology.
result Adding topological features to GCN significantly improves accuracy over state-of-the-art methods.
A new method for graph-structured data improves transformer performance by incorporating topology.
problem Improving transformer performance on graph-structured data.
method Parameterizing topological masks as a learnable function of a weighted adjacency matrix, approximated with graph random features.
result Efficient masking algorithms provide strong performance gains for tasks on image and point cloud data.
A new model learns graph structures from data.
problem Learning graph topologies from data.
method Proposes a learning to optimise (L2O) approach to learn graph structures from node data.
result The proposed model learns graph structures more efficiently than classic iterative algorithms.
Proposes a graph pooling method leveraging node proximity for hierarchical graph representation learning.
problem Efficiently exploiting the geometry of graph data for hierarchical representation learning.
method Combines node proximity with kernel representation of topology and node features for adaptive node signal similarities evaluation.
result Achieves state-of-the-art performance on graph classification benchmark datasets.
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.