# Mixed-Order Spectral Clustering for Networks

## TL;DR

This paper introduces a novel mixed-order spectral clustering method that simultaneously models second- and third-order network structures, improving clustering accuracy and providing new evaluation metrics.

## Contribution

It proposes two MOSC methods based on Graph Laplacian and Random Walks, with automatic mixing parameter selection and structure-aware error metrics.

## Key findings

- MOSC methods outperform existing spectral clustering approaches
- Automatic mixing parameter determination is effective
- New error metrics offer valuable insights

## Abstract

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.