Study orders of canonical bundles over graph configuration spaces.
problem Determining bundle orders for planar and nonplanar graphs.
method Analyzing configuration spaces of graphs to find bundle orders.
result Bundle orders are 2 for planar and 4 for nonplanar graphs.
We present novel graph kernels for graphs with node and edge labels that have ordered neighborhoods, i.e. when neighbor nodes follow an order. Graphs with ordered neighborhoods are a natural data representation for evolving graphs where edges are created over time, which induces an order. Combining convolutional subgra…
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar graphs.
kth-order invariant graph networks are as powerful as kth-order WL in distinguishing graphs.
problem Measuring the expressive power of graph neural network formalisms.
method Considered kth-order invariant graph networks (k-IGNs) and compared their expressive power to kth-order WL.
result k-IGNs and k-WL are equally powerful in distinguishing graphs.
Study finds maximal linklessly embeddable graphs up to 11 vertices and their complements.
problem Characterizing linklessly embeddable graphs and their complements.
method Comprehensive search and verification of graphs up to 11 vertices.
result For graphs of order 11, either the graph or its complement is intrinsically linked.
The complement of a non-separating planar graph contains a K_n minor.
problem Characterizing the structure of complements of planar graphs.
method Analyzing the structure of complements of non-separating planar graphs and using examples to illustrate hypotheses.
result The order 2n-3 is the lowest possible for a non-separating planar graph whose complement contains a K_n minor.
PARD generates graphs efficiently and invariantly to node ordering.
problem Graph generation sensitivity to node ordering.
method Integrates autoregressive and diffusion models with a partial order for nodes and edges.
result PARD achieves state-of-the-art performance on molecular and non-molecular datasets.
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.
Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.
problem Tackles the limitation of previous equivariant neural networks by considering permutations of matrices.
method Derives formulae for general permutation equivariant layers, including matrix permutations. Presents a second order graph variational encoder.
result Latent distribution of equivariant generative models must be exchangeable.
A new autoregressive model learns the order of graph generation tasks.
problem Generating graphs in a meaningful order when the canonical order is not obvious.
method Introduces a variant of autoregressive models that dynamically decides the autoregressive order based on data.
result Achieves state-of-the-art results on molecular graph generation benchmarks.
New model generates graphs with tighter likelihood bounds and better quality.
problem Intractable likelihood of autoregressive graph models.
method Derive exact joint probability, approximate node orderings, variational inference.
result Lower bound on log-likelihood is significantly tighter than previous methods.
Study compares hypergraph and graph-level models for higher-order relational learning.
problem Evaluating effectiveness of hypergraph-level vs. graph-level models in relational learning.
method Systematic evaluation of various hypergraph and graph-level architectures.
result Graph-level models applied to hypergraph expansions outperform hypergraph-level models.
Graphs and their complements are intrinsically knotted.
problem Characterizing maximal linklessly embeddable graphs and their complements.
method Analyzing maximal linklessly embeddable graphs, deriving connected domination numbers, and proving intrinsic knotting properties.
result Complements of maximal linklessly embeddable graphs of order 12 and 15 are intrinsically knotted.
With higher-order neighborhood information of graph network, the accuracy of graph representation learning classification can be significantly improved. However, the current higher order graph convolutional network has a large number of parameters and high computational complexity. Therefore, we propose a Hybrid Lower …
A new method predicts higher-order interactions in evolving graphs using simplicial complexes.
problem Predicting higher-order interactions in dynamic graphs with theoretical guarantees.
method Capturing higher-order interactions as simplices, modeling neighborhoods with face-vectors, and developing a nonparametric kernel estimator.
result Our method outperforms existing higher-order prediction methods and is theoretically consistent.
Directed graphs occur throughout statistical modeling of networks, and exchangeability is a natural assumption when the ordering of vertices does not matter. There is a deep structural theory for exchangeable undirected graphs, which extends to the directed case via measurable objects known as digraphons. Using digraph…
Combines neural networks and probabilistic graphical models for efficient higher-order inference.
problem Lack of efficient higher-order relational information in graph neural networks and probabilistic graphical models.
method Derives efficient approximate sum-product loopy belief propagation for higher-order PGMs, embeds into neural network, proposes methods for constructing higher-order factors.
result Substantially outperforms state-of-the-art k-order graph neural networks in molecular datasets.
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
Attributed graph clustering is challenging as it requires joint modelling of graph structures and node attributes. Recent progress on graph convolutional networks has proved that graph convolution is effective in combining structural and content information, and several recent methods based on it have achieved promisin…
In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical…
Graph neural network have achieved impressive results in predicting molecular properties, but they do not directly account for local and hidden structures in the graph such as functional groups and molecular geometry. At each propagation step, GNNs aggregate only over first order neighbours, ignoring important informat…
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.
We prove that every simple graph of order 12 which has minimum degree 6 contains a K_6 minor.
Graph embedding has become a key component of many data mining and analysis systems. Current graph embedding approaches either sample a large number of node pairs from a graph to learn node embeddings via stochastic optimization or factorize a high-order proximity/adjacency matrix of the graph via computationally expen…
We introduce nonlinear higher-order label spreading for semi-supervised learning.
problem Efficient semi-supervised learning on graphs with complex label spreading.
method We add nonlinearity to label spreading through higher-order graph structures, proving convergence and demonstrating efficiency on various datasets.
result Our nonlinear higher-order label spreading algorithm converges to the global solution and performs favorably compared to classical methods.
Proposes a 2-WL-based graph convolution for improved graph classification.
problem Limitations of current GNN architectures in discriminative power.
method Introduces a novel 2-dimensional Weisfeiler-Lehman graph convolution.
result 2-WL-GNN architecture is more discriminative than existing GNN approaches.
Graph-based methods are known to be successful in many machine learning and pattern classification tasks. These methods consider semi-structured data as graphs where nodes correspond to primitives (parts, interest points, segments, etc.) and edges characterize the relationships between these primitives. However, these …
New approach learns graph representations by contrasting first-order neighbors and graph diffusion views.
problem Learning node and graph level representations from graph data.
method Self-supervised approach using contrastive learning of multi-scale encodings.
result Achieves state-of-the-art performance on 8 out of 8 benchmarks.
FairGP uses graph partitioning to make Graph Transformers fair and scalable.
problem Fairness issues in Graph Transformers, especially against sensitive features.
method Graph partitioning to minimize the influence of higher-order nodes and optimize attention mechanisms.
result FairGP improves fairness in Graph Transformers while reducing computational complexity.
SASE improves attributed graph clustering for large graphs with linear time and space complexity.
problem Challenges in clustering large attributed graphs due to high computational and memory costs.
method SASE combines node features smoothing, scalable spectral clustering, and adaptive order selection.
result SASE achieves a 6.9% improvement in ACC and a 5.87x speedup on the ArXiv dataset.
In this paper, we presented a novel convolutional neural network framework for graph modeling, with the introduction of two new modules specially designed for graph-structured data: the k-th order convolution operator and the adaptive filtering module. Importantly, our framework of High-order and Adaptive Graph Convo…
This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significa…
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L, there is an associated ribbon graph whose quasi-trees correspond bijectively to …
Predicts node sequences in graphs using multi-order network models.
problem Predicting sequences of node traversals in graphs.
method Combines multiple higher-order network models into a multi-order model, fitting and selecting the optimal maximum order.
result Outperforms state-of-the-art algorithms for next-element and full sequence prediction.
New method for learning on heterogeneous graphs without meta-paths.
problem Learning on heterogeneous graphs is sensitive to meta-paths choice, leading to poor performance.
method Decompose heterogeneous graph into homogeneous relation-type graphs, combine higher-order representations, use attention mechanisms.
result Our model outperforms state-of-the-art baselines in vertex classification tasks on heterogeneous graph datasets.
Proposes an approach to ensure acyclic graphs in Bayesian structure learning.
problem Ensuring acyclic graphs in Bayesian structure learning.
method Integration of knowledge from topological orderings to constrain acyclicty.
result Outperforms related Bayesian score-based approaches in experiments.
In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…
Novel CG-EGNNs learn equivariant functions from Clifford algebras.
problem Lack of equivariance in high-order graph neural networks.
method Integrates high-order local structures with Clifford algebras for equivariant learning.
result CG-EGNNs outperform previous methods on various benchmarks.
Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations…
We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for k≥18, most graphs of order k are intrinsically knotted and, for k≥2n+9, most of order k are not n-apex…
Graph Convolutional Networks (GCNs) have proven to be successful tools for semi-supervised learning on graph-based datasets. For sparse graphs, linear and polynomial filter functions have yielded impressive results. For large non-sparse graphs, however, network training and evaluation becomes prohibitively expensive. B…
Graph Convolution Network (GCN) has been recognized as one of the most effective graph models for semi-supervised learning, but it extracts merely the first-order or few-order neighborhood information through information propagation, which suffers performance drop-off for deeper structure. Existing approaches that deal…
Paper proposes methods to learn DAGs from partial orderings.
problem Learning DAGs from partial orderings is challenging.
method General estimation framework and efficient algorithms for low- and high-dimensional problems.
result Efficient estimation of DAGs from partial orderings is possible.
The paper studies the graph geometry of finite groups, creating a dataset and analyzing its properties.
problem Understanding how group-theoretic structure is reflected in Cayley graph observables.
method Construction of a dataset of Cayley graphs for groups of order up to 767, analysis of graph statistics, and comparison of model performance.
result Graph statistics are highly informative for predicting group properties, and GNNs can recover substantial structural signal.
H-GAT improves stock selection by capturing complex higher-order stock relations and integrating both technical and fundamental analysis.
problem Stock selection difficulty and lack of comprehensive analysis.
method Higher-order Graph Attention Network (H-GAT) that incorporates both technical and fundamental analysis.
result H-GAT outperforms existing methods in stock selection metrics.
BScNets expands graph learning to higher-order interactions.
problem Link prediction on graphs with higher-order interactions.
method Proposes a new SNN using block Hodge-Laplacian and block simplicial complexes.
result BScNets outperforms state-of-the-art models in link prediction.
We consider the orientation-preserving actions of finite groups G on pairs (S3,Γ), where Γ is a connected graph of genus g>1, embedded in S3. For each g we give the maximum order mg of such G acting on (S3,Γ) for all such Γ⊂S3. Indeed we will classify all graphs Γ⊂S3 which re…