# Complex non-backtracking matrix for directed graphs

arXiv: https://arxiv.org/abs/2507.12503 · HTML: https://mlsanity.com/paper/2507.12503/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 16 Jul 2025.

## Builds on (most cited first, 11 of 11)

- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Spectral Clustering of Graphs with the Bethe Hessian](https://mlsanity.com/paper/1406.1880/) (2014)
- [Higher-Order Spectral Clustering of Directed Graphs](https://mlsanity.com/paper/2011.05080/) (2020)
- [Localization and centrality in networks](https://mlsanity.com/paper/1401.5093/) (2014)
- [Spectral community detection in sparse networks](https://mlsanity.com/paper/1308.6494/) (2013)
- [Skew-Symmetric Adjacency Matrices for Clustering Directed Graphs](https://mlsanity.com/paper/2203.01388/) (2022)
- [A message-passing approach for recurrent-state epidemic models on networks](https://mlsanity.com/paper/1505.02192/) (2015)
- [Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models](https://mlsanity.com/paper/1609.02487/) (2016)
- [An iterative spectral algorithm for digraph clustering](https://mlsanity.com/paper/2501.17951/) (2025)
- [Leveraging percolation theory to single out influential spreaders in networks](https://mlsanity.com/paper/1605.07041/) (2016)

## Abstract (arXiv)

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.