# Haar-Laplacian for directed graphs

arXiv: https://arxiv.org/abs/2411.15527 · HTML: https://mlsanity.com/paper/2411.15527/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 56 references that are in the map.
- First posted on arXiv 23 Nov 2024.

## Builds on (most cited first, 24 of 24)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [Directed Graph Convolutional Network](https://mlsanity.com/paper/2004.13970/) (2020)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [SigMaNet: One Laplacian to Rule Them All](https://mlsanity.com/paper/2205.13459/) (2022)
- [SSSNET: Semi-Supervised Signed Network Clustering](https://mlsanity.com/paper/2110.06623/) (2021)
- [Signed Graph Neural Networks: A Frequency Perspective](https://mlsanity.com/paper/2208.07323/) (2022)
- [Irregularity-Aware Graph Fourier Transforms](https://mlsanity.com/paper/1802.10220/) (2018)
- [wsGAT: Weighted and Signed Graph Attention Networks for Link Prediction](https://mlsanity.com/paper/2109.11519/) (2021)
- [MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/2110.07570/) (2021)
- [Line Graph Neural Networks for Link Weight Prediction](https://mlsanity.com/paper/2309.15728/) (2023)
- [DUPLEX: Dual GAT for Complex Embedding of Directed Graphs](https://mlsanity.com/paper/2406.05391/) (2024)
- [A Directed Graph Fourier Transform with Spread Frequency Components](https://mlsanity.com/paper/1804.03000/) (2018)
- [Graph Fourier transform based on singular value decomposition of directed Laplacian](https://mlsanity.com/paper/2205.06242/) (2022)
- [Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs](https://mlsanity.com/paper/2312.17361/) (2023)
- [Predicting missing links and their weights via reliable-route-based method](https://mlsanity.com/paper/1408.0845/) (2014)
- [Frequency analysis and filter design for directed graphs with polar decomposition](https://mlsanity.com/paper/2312.11421/) (2023)
- [NEW: A Generic Learning Model for Tie Strength Prediction in Networks](https://mlsanity.com/paper/2001.05283/) (2020)

## Built on by (most cited first, 1 of 1)

- [Mag-Mamba: Modeling Coupled spatiotemporal Asymmetry for POI Recommendation](https://mlsanity.com/paper/2603.00053/) (2026)

## Abstract (arXiv)

This paper introduces a novel Laplacian matrix aiming to enable the construction of spectral convolutional networks and to extend the signal processing applications for directed graphs. Our proposal is inspired by a Haar-like transformation and produces a Hermitian matrix which is not only in one-to-one relation with the adjacency matrix, preserving both direction and weight information, but also enjoys desirable additional properties like scaling robustness, sensitivity, continuity, and directionality. We take a theoretical standpoint and support the conformity of our approach with the spectral graph theory. Then, we address two use-cases: graph learning (by introducing HaarNet, a spectral graph convolutional network built with our Haar-Laplacian) and graph signal processing. We show that our approach gives better results in applications like weight prediction and denoising on directed graphs.