  Clustering is fundamental for gaining insights from complex networks, and
spectral clustering (SC) is a popular approach. Conventional SC focuses on
second-order structures (e.g., edges connecting two nodes) without direct
consideration of higher-order structures (e.g., triangles and cliques). This
has motivated SC extensions that directly consider higher-order structures.
However, both approaches are limited to considering a single order. This paper
proposes a new Mixed-Order Spectral Clustering (MOSC) approach to model both
second-order and third-order structures simultaneously, with two MOSC methods
developed 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. We automatically determine the mixing parameter based on cut
criteria or triangle density, and construct new structure-aware error metrics
for performance evaluation. Experiments on real-world networks show 1) the
superior performance of two MOSC methods over existing SC methods, 2) the
effectiveness of the mixing parameter determination strategy, and 3) insights
offered by the structure-aware error metrics.
