Cycles are fundamental elements in graph-structured data and have demonstrated their effectiveness in enhancing graph learning models. To encode such information into a graph learning framework, prior works often extract a summary quantity, ranging from the number of cycles to the more sophisticated persistence diagram summaries. However, more detailed information, such as which edges are encoded in a cycle, has not yet been used in graph neural networks. In this paper, we make one step towards addressing this gap, and propose a structure encoding module, called CycleNet, that encodes cycle information via edge structure encoding in a permutation invariant manner. To efficiently encode the space of all cycles, we start with a cycle basis (i.e., a minimal set of cycles generating the cycle space) which we compute via the kernel of the 1-dimensional Hodge Laplacian of the input graph. To guarantee the encoding is invariant w.r.t. the choice of cycle basis, we encode the cycle information via the orthogonal projector of the cycle basis, which is inspired by BasisNet proposed by Lim et al. We also develop a more efficient variant which however requires that the input graph has a unique shortest cycle basis. To demonstrate the effectiveness of the proposed module, we provide some theoretical understandings of its expressive power. Moreover, we show via a range of experiments that networks enhanced by our CycleNet module perform better in various benchmarks compared to several existing SOTA models.

翻译：循环是图结构数据中的基本元素，已被证明能有效提升图学习模型性能。为将此类信息编码到图学习框架中，现有工作通常提取汇总量（从循环数量到更复杂的持续图摘要）。然而，诸如循环编码了哪些边等更细节的信息尚未在图神经网络中得到应用。本文向解决这一差距迈进一步，提出名为CycleNet的结构编码模块，通过置换不变的方式利用边结构编码对循环信息进行编码。为高效编码所有循环构成的向量空间，我们从循环基（即生成循环空间的最小循环集）出发，通过输入图的1维霍奇拉普拉斯矩阵的核计算该基。为确保编码对循环基的选择具有不变性，我们借鉴Lim等人提出的BasisNet思想，通过循环基的正交投影算子对循环信息进行编码。我们还开发了一种更高效的变体，其前提条件是输入图需具有唯一的最短循环基。为证明所提模块的有效性，我们对其表达能力进行了理论分析。此外，通过一系列实验表明，与现有多种SOTA模型相比，采用CycleNet模块增强的网络在多种基准测试中表现更优。