Spectral clustering (SC) is a popular approach for gaining insights from complex networks. Conventional SC focuses on second-order structures (e.g. edges) without direct consideration of higher-order structures (e.g. triangles). This has motivated SC extensions that directly consider higher-order structures. However, both approaches are limited to considering a single order. To address this issue, this paper proposes a novel Mixed-Order Spectral Clustering (MOSC) framework to model both second-order and third-order structures simultaneously. To model mixed-order structures, we propose two new methods based on Graph Laplacian (GL) and Random Walks (RW). MOSC-GL combines edge and triangle adjacency matrices, with theoretical performance guarantee. MOSC-RW combines first-order and second-order random walks for a probabilistic interpretation. Moreover, we design mixed-order cut criteria to enable existing SC methods to preserve mixed-order structures, and develop new mixed-order evaluation metrics for structure-level evaluation. Experiments on community detection and superpixel segmentation show (1) the superior performance of the MOSC methods over existing SC methods, (2) enhanced performance of conventional SC due to mixed-order cut criteria, and (3) new insights of output clusters offered by the mixed-order evaluation metrics.

频谱聚类（SC）是一种从复杂网络中获取洞见的流行方法。常规的SC关注于二阶结构（例如，边缘），而没有直接考虑高阶结构（例如，三角形）。这激发了直接考虑高阶结构的SC扩展。但是，这两种方法都限于考虑单个订单。为了解决这个问题，本文提出了一种新颖的混合顺序频谱聚类（MOSC）框架可同时对二阶和三阶结构进行建模。为了对混合顺序结构建模，我们提出了两种基于图拉普拉斯算子（GL）和随机游走（RW）的新方法。MOSC-GL结合了边缘和三角形邻接矩阵，并具有理论上的性能保证。MOSC-RW结合了一阶和二阶随机游走以进行概率解释。此外，我们设计了混合顺序切割标准，以使现有的SC方法能够保留混合顺序结构，并开发新的混合顺序评估指标以进行结构级评估。社区检测和超像素分割的实验表明：（1）MOSC方法优于现有SC方法的性能；（2）由于混合顺序切割标准，传统SC的性能得到了提高；