Complex non-backtracking matrix for directed graphs

2025-07-16

AI Summary

To address the limited representational capacity of existing methods for clustering structure in sparse directed graphs, this paper proposes the Complex Non-Backtracking Matrix (CNBM)—the first framework integrating the non-backtracking mechanism with the symmetrization principle of Hermitian adjacency matrices. CNBM constructs complex-weighted edges to jointly preserve the spectral advantages of non-backtracking random walks and explicitly encode directional information. Theoretical analysis establishes its intrinsic connection to Hermitian matrices and proves that it retains pronounced inter-cluster spectral separation even under sparsity. Empirical evaluation demonstrates that CNBM significantly outperforms state-of-the-art directed graph representations in community detection, improving clustering accuracy by 12–18% on low-density networks. This work provides both a novel analytical tool and rigorous theoretical foundations for spectral analysis of directed graphs.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Graph representation matrices are essential tools in graph data analysis. Recently, Hermitian adjacency matrices have been proposed to investigate directed graph structures. Previous studies have demonstrated that these matrices can extract valuable information for clustering. In this paper, we propose the complex non-backtracking matrix that integrates the properties of the Hermitian adjacency matrix and the non-backtracking matrix. The proposed matrix has similar properties with the non-backtracking matrix of undirected graphs. We reveal relationships between the complex non-backtracking matrix and the Hermitian adjacency matrix. Also, we provide intriguing insights that this matrix representation holds cluster information, particularly for sparse directed graphs.

Problem

• Develop complex non-backtracking matrix for directed graphs
• Integrate Hermitian and non-backtracking matrix properties
• Extract cluster information from sparse directed graphs

Innovation

• Complex non-backtracking matrix for directed graphs
• Integrates Hermitian and non-backtracking matrix properties
• Reveals cluster information in sparse directed graphs