Neural Information Processing Systems http://nips.cc/

Modeling long-range interactions, the propagation of information across distant parts of a graph, is a central challenge in graph machine learning. Graph wavelets, inspired by multi-resolution signal processing, provide a principled way to capture both local and global structures. However, existing wavelet-based graph neural networks rely on finite-order polynomial approximations, which limit their receptive fields and hinder long-range propagation. We propose Long-Range Graph Wavelet Networks (LR-GWN), which decompose wavelet filters into complementary local and global components. Local aggregation is handled with efficient low-order polynomials, while long-range interactions are captured through a flexible spectral-domain parameterization. This hybrid design unifies short- and long-distance information flow within a principled wavelet framework. Experiments show that LR-GWN achieves state-of-the-art performance among wavelet-based methods on long-range benchmarks, while remaining competitive on short-range datasets.

L and use the binomial equation, i.e. To obtain a more convenient form for K we shift the summation index using m = k j, i.e. ξ All remaining nodes are part of the test set and only used once for testing. Different seeds are used for validation and test splits. The patience is reset after an increase in accuracy on the validation set. We use the same development set for optimizing the hyperparameters for clustering.

Neural Information Processing Systems http://nips.cc/

Code is available at https://github.com/ivam-he/BernNet .

Graph Neural Networks (GNNs) have achieved significant success across various domains by leveraging graph structures in data. Existing spectral GNNs, which use low-degree polynomial filters to capture graph spectral properties, may not fully identify the graph's spectral characteristics because of the polynomial's small degree. However, increasing the polynomial degree is computationally expensive and beyond certain thresholds leads to performance plateaus or degradation. In this paper, we introduce the Piecewise Constant Spectral Graph Neural Network(PieCoN) to address these challenges. PieCoN combines constant spectral filters with polynomial filters to provide a more flexible way to leverage the graph structure. By adaptively partitioning the spectrum into intervals, our approach increases the range of spectral properties that can be effectively learned. Experiments on nine benchmark datasets, including both homophilic and heterophilic graphs, demonstrate that PieCoN is particularly effective on heterophilic datasets, highlighting its potential for a wide range of applications.

Message passing neural networks (MPNNs) have been shown to have limitations in terms of expressivity and modeling long-range interactions. Augmenting MPNNs with a virtual node (VN) removes the locality constraint of the layer aggregation and has been found to improve performance on a range of benchmarks. We provide a comprehensive theoretical analysis of the role of VNs and benefits thereof, through the lenses of oversmoothing, oversquashing, and sensitivity analysis. First, in contrast to prior belief, we find that VNs typically avoid replicating anti-smoothing approaches to maintain expressive power. Second, we characterize, precisely, how the improvement afforded by VNs on the mixing abilities of the network and hence in mitigating oversquashing, depends on the underlying topology. Finally, we highlight that, unlike Graph-Transformers (GT), classical instantiations of the VN are often constrained to assign uniform importance to different nodes. Consequently, we propose a variant of VN with the same computational complexity, which can have different sensitivity to nodes based on the graph structure. We show that this is an extremely effective and computationally efficient baseline on graph-level tasks.

Graph Neural Networks (GNNs), known as spectral graph filters, find a wide range of applications in web networks. To bypass eigendecomposition, polynomial graph filters are proposed to approximate graph filters by leveraging various polynomial bases for filter training. However, no existing studies have explored the diverse polynomial graph filters from a unified perspective for optimization. In this paper, we first unify polynomial graph filters, as well as the optimal filters of identical degrees into the Krylov subspace of the same order, thus providing equivalent expressive power theoretically. Next, we investigate the asymptotic convergence property of polynomials from the unified Krylov subspace perspective, revealing their limited adaptability in graphs with varying heterophily degrees. Inspired by those facts, we design a novel adaptive Krylov subspace approach to optimize polynomial bases with provable controllability over the graph spectrum so as to adapt various heterophily graphs. Subsequently, we propose AdaptKry, an optimized polynomial graph filter utilizing bases from the adaptive Krylov subspaces. Meanwhile, in light of the diverse spectral properties of complex graphs, we extend AdaptKry by leveraging multiple adaptive Krylov bases without incurring extra training costs. As a consequence, extended AdaptKry is able to capture the intricate characteristics of graphs and provide insights into their inherent complexity. We conduct extensive experiments across a series of real-world datasets. The experimental results demonstrate the superior filtering capability of AdaptKry, as well as the optimized efficacy of the adaptive Krylov basis.

Spectral Graph Neural Networks (GNNs), alternatively known as graph filters, have gained increasing prevalence for heterophily graphs. Optimal graph filters rely on Laplacian eigendecomposition for Fourier transform. In an attempt to avert prohibitive computations, numerous polynomial filters have been proposed. However, polynomials in the majority of these filters are predefined and remain fixed across different graphs, failing to accommodate the varying degrees of heterophily. Addressing this gap, we demystify the intrinsic correlation between the spectral property of desired polynomial bases and the heterophily degrees via thorough theoretical analyses. Subsequently, we develop a novel adaptive heterophily basis wherein the basis vectors mutually form angles reflecting the heterophily degree of the graph. We integrate this heterophily basis with the homophily basis to construct a universal polynomial basis UniBasis, which devises a polynomial filter based graph neural network - UniFilter. It optimizes the convolution and propagation in GNN, thus effectively limiting over-smoothing and alleviating over-squashing. Our extensive experiments, conducted on a diverse range of real-world and synthetic datasets with varying degrees of heterophily, support the superiority of UniFilter. These results not only demonstrate the universality of UniBasis but also highlight its proficiency in graph explanation.