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Multi-order Graph Clustering with Adaptive Node-level Weight Learning

Ye Liu, Xuelei Lin, Yejia Chen, Reynold Cheng

TL;DR

The paper tackles fragmentation in motif-based hypergraphs and the challenge of integrating multiple motifs for graph clustering. It introduces MOGC, a framework that learns per-node weights over multiple motifs and edges to produce a fused adjacency suitable for spectral clustering, optimized via alternating minimization with a regularization on motif weights. Key contributions include adaptive node-level motif weighting, a formal fused-adjacency construction, and theoretical convergence with complexity analysis, validated on seven real datasets where MOGC outperforms single-motif and baseline methods. The approach offers a practical path to leveraging higher-order structure in real networks and sets the stage for nonlinear fusion and GNN-based enhancements.

Abstract

Current graph clustering methods emphasize individual node and edge con nections, while ignoring higher-order organization at the level of motif. Re cently, higher-order graph clustering approaches have been designed by motif based hypergraphs. However, these approaches often suffer from hypergraph fragmentation issue seriously, which degrades the clustering performance greatly. Moreover, real-world graphs usually contain diverse motifs, with nodes participating in multiple motifs. A key challenge is how to achieve precise clustering results by integrating information from multiple motifs at the node level. In this paper, we propose a multi-order graph clustering model (MOGC) to integrate multiple higher-order structures and edge connections at node level. MOGC employs an adaptive weight learning mechanism to au tomatically adjust the contributions of different motifs for each node. This not only tackles hypergraph fragmentation issue but enhances clustering accuracy. MOGC is efficiently solved by an alternating minimization algo rithm. Experiments on seven real-world datasets illustrate the effectiveness of MOGC.

Multi-order Graph Clustering with Adaptive Node-level Weight Learning

TL;DR

The paper tackles fragmentation in motif-based hypergraphs and the challenge of integrating multiple motifs for graph clustering. It introduces MOGC, a framework that learns per-node weights over multiple motifs and edges to produce a fused adjacency suitable for spectral clustering, optimized via alternating minimization with a regularization on motif weights. Key contributions include adaptive node-level motif weighting, a formal fused-adjacency construction, and theoretical convergence with complexity analysis, validated on seven real datasets where MOGC outperforms single-motif and baseline methods. The approach offers a practical path to leveraging higher-order structure in real networks and sets the stage for nonlinear fusion and GNN-based enhancements.

Abstract

Current graph clustering methods emphasize individual node and edge con nections, while ignoring higher-order organization at the level of motif. Re cently, higher-order graph clustering approaches have been designed by motif based hypergraphs. However, these approaches often suffer from hypergraph fragmentation issue seriously, which degrades the clustering performance greatly. Moreover, real-world graphs usually contain diverse motifs, with nodes participating in multiple motifs. A key challenge is how to achieve precise clustering results by integrating information from multiple motifs at the node level. In this paper, we propose a multi-order graph clustering model (MOGC) to integrate multiple higher-order structures and edge connections at node level. MOGC employs an adaptive weight learning mechanism to au tomatically adjust the contributions of different motifs for each node. This not only tackles hypergraph fragmentation issue but enhances clustering accuracy. MOGC is efficiently solved by an alternating minimization algo rithm. Experiments on seven real-world datasets illustrate the effectiveness of MOGC.
Paper Structure (27 sections, 19 equations, 7 figures, 5 tables, 1 algorithm)

This paper contains 27 sections, 19 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Lower-order Graph Clustering
  4. Motif-based Graph Clustering
  5. Preliminaries
  6. Network Motif
  7. Motif Adjacency matrix
  8. 3-node Motif-based Adjacency Matrix Construction
  9. 4-node and 5-node Motif-based Adjacency Matrix Construction
  10. The Proposed Method
  11. Optimization
  12. Fix $\mathbf{\Lambda}$, solve $\mathbf{U}$:
  13. Fix $\mathbf{U}$, solve $\mathbf{\Lambda}$:
  14. Theoretical Analysis
  15. Convergence Analysis
  16. ...and 12 more sections

Figures (7)

  • Figure 1: Illustration of fragmentation issue and different motif's contributions to the graph clustering task: (a) Original network is fragmented into one connected component (blue nodes) and six isolated nodes (red nodes and green node) based on 3-node motif. (b) Original network is fragmented into one connected component (red nodes) and six isolated nodes (blue nodes and green node) based on 4-node motif. Original network contains edges with weight 1, numbers on edges demonstrate the edge weight in 3-node motif based hypergraph and 4-node motif based hypergraph, the corresponding motif adjacency matrix is shown in the third column.
  • Figure 2: Network motif example for 3-node motifs. ${M}_3^3$ and ${M}_3^2$ are simple motifs, since the anchor set $\mathcal{A}$ contains all of the nodes in ${M}_3^3$ and ${M}_3^2$. $\mathbf{B}$ is the binary matrix encoding edge pattern of corresponding motif.
  • Figure 3: The flowchart of MOGC
  • Figure 4: 4-node motif and 5-node motif used in the experiments.
  • Figure 5: The weight in adjacency matrix for nodes on polbooks dataset. The red dash lines are used to separate the nodes from different clusters.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • proof 1