# Skew-Symmetric Adjacency Matrices for Clustering Directed Graphs

arXiv: https://arxiv.org/abs/2203.01388 · HTML: https://mlsanity.com/paper/2203.01388/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 8 of 8)

- [Higher-order organization of complex networks](https://mlsanity.com/paper/1612.08447/) (2016)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Clustering and Community Detection in Directed Networks: A Survey](https://mlsanity.com/paper/1308.0971/) (2013)
- [The Impact of Random Models on Clustering Similarity](https://mlsanity.com/paper/1701.06508/) (2017)
- [Inhomogeneous Hypergraph Clustering with Applications](https://mlsanity.com/paper/1709.01249/) (2017)
- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Higher-Order Spectral Clustering of Directed Graphs](https://mlsanity.com/paper/2011.05080/) (2020)
- [Spectral clustering algorithms for the detection of clusters in block-cyclic and block-acyclic graphs](https://mlsanity.com/paper/1805.00862/) (2018)

## Built on by (most cited first, 4 of 4)

- [Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs](https://mlsanity.com/paper/2203.03221/) (2022)
- [Spectral Clustering for Directed Graphs via Likelihood Estimation on Stochastic Block Models](https://mlsanity.com/paper/2403.19516/) (2024)
- [Complex non-backtracking matrix for directed graphs](https://mlsanity.com/paper/2507.12503/) (2025)
- [Hierarchical Clustering and Signal Denoising on Digraphs](https://mlsanity.com/paper/2609.34670/) (2026)

## Abstract (arXiv)

Cut-based directed graph (digraph) clustering often focuses on finding dense within-cluster or sparse between-cluster connections, similar to cut-based undirected graph clustering methods. In contrast, for flow-based clusterings the edges between clusters tend to be oriented in one direction and have been found in migration data, food webs, and trade data. In this paper we introduce a spectral algorithm for finding flow-based clusterings. The proposed algorithm is based on recent work which uses complex-valued Hermitian matrices to represent digraphs. By establishing an algebraic relationship between a complex-valued Hermitian representation and an associated real-valued, skew-symmetric matrix the proposed algorithm produces clusterings while remaining completely in the real field. Our algorithm uses less memory and asymptotically less computation while provably preserving solution quality. We also show the algorithm can be easily implemented using standard computational building blocks, possesses better numerical properties, and loans itself to a natural interpretation via an objective function relaxation argument.