Complex non-backtracking matrix for directed graphs

TL;DR

This paper introduces a complex non-backtracking matrix for directed graphs that combines properties of Hermitian adjacency and non-backtracking matrices, revealing its effectiveness in extracting cluster information, especially in sparse directed graphs.

Contribution

The paper proposes a novel complex non-backtracking matrix for directed graphs, linking it to existing matrices and demonstrating its ability to capture clustering structure.

Findings

The matrix relates to Hermitian adjacency matrices.

It effectively captures cluster information in sparse directed graphs.

The matrix generalizes non-backtracking matrices to directed graphs.

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.