Spectral Graph Neural Networks (GNNs), also referred to 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 the prohibitive computations, numerous polynomial filters by leveraging distinct polynomials have been proposed to approximate the desired graph filters. However, polynomials in the majority of polynomial filters are predefined and remain fixed across all graphs, failing to accommodate the diverse heterophily degrees across different graphs. To tackle this issue, we first investigate the correlation between polynomial bases of desired graph filters and the degrees of graph heterophily via a thorough theoretical analysis. Afterward, we develop an adaptive heterophily basis by incorporating graph heterophily degrees. Subsequently, we integrate this heterophily basis with the homophily basis, creating a universal polynomial basis UniBasis. In consequence, we devise a general polynomial filter UniFilter. Comprehensive experiments on both real-world and synthetic datasets with varying heterophily degrees significantly support the superiority of UniFilter, demonstrating the effectiveness and generality of UniBasis, as well as its promising capability as a new method for graph analysis.

翻译：谱图神经网络（Spectral GNNs，也称图滤波器）在异配图中的应用日益广泛。最优图滤波器依赖于拉普拉斯特征分解进行傅里叶变换。为避免高昂的计算成本，研究者利用多种不同多项式提出了众多多项式滤波器，以逼近所需的图滤波器。然而，大多数多项式滤波器中的多项式是预先定义的，且在所有图上保持不变，无法适应不同图之间异配程度的差异。为解决这一问题，我们首先通过深入的理论分析，研究了所需图滤波器多项式基与图异配程度之间的相关性。随后，我们通过融入图异配程度，开发了一种自适应异配基。接着，我们将此异配基与同配基相结合，构建了一个通用多项式基UniBasis。基于此，我们设计了一种通用多项式滤波器UniFilter。在具有不同异配程度的真实和合成数据集上进行的综合实验显著支持了UniFilter的优越性，证明了UniBasis的有效性和通用性，以及其作为图分析新方法的巨大潜力。