# MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs

arXiv: https://arxiv.org/abs/2110.07570 · HTML: https://mlsanity.com/paper/2110.07570/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 46 references that are in the map.
- First posted on arXiv 14 Oct 2021.

## Builds on (most cited first, 21 of 21)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Break the Ceiling: Stronger Multi-scale Deep Graph Convolutional Networks](https://mlsanity.com/paper/1906.02174/) (2019)
- [PageRank beyond the Web](https://mlsanity.com/paper/1407.5107/) (2014)
- [Spectral-based Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/1907.08990/) (2019)
- [Autoregressive Moving Average Graph Filtering](https://mlsanity.com/paper/1602.04436/) (2016)
- [Optimizing Generalized PageRank Methods for Seed-Expansion Community Detection](https://mlsanity.com/paper/1905.10881/) (2019)
- [Vertex-Frequency Analysis on Graphs](https://mlsanity.com/paper/1307.5708/) (2013)
- [Magnetic Eigenmaps for the Visualization of Directed Networks](https://mlsanity.com/paper/1606.08266/) (2016)
- [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016)
- [Design of graph filters and filterbanks](https://mlsanity.com/paper/1711.02046/) (2017)
- [Complex Networks from Classical to Quantum](https://mlsanity.com/paper/1702.08459/) (2017)
- [Dimensional Reweighting Graph Convolutional Networks](https://mlsanity.com/paper/1907.02237/) (2019)

## Built on by (most cited first, 7 of 7)

- [Breaking the Entanglement of Homophily and Heterophily in Semi-supervised Node Classification](https://mlsanity.com/paper/2312.04111/) (2023)
- [LightDiC: A Simple yet Effective Approach for Large-scale Digraph Representation Learning](https://mlsanity.com/paper/2401.11772/) (2024)
- [DiRW: Path-Aware Digraph Learning for Heterophily](https://mlsanity.com/paper/2410.10320/) (2024)
- [Towards Data-centric Machine Learning on Directed Graphs: a Survey](https://mlsanity.com/paper/2412.01849/) (2024)
- [Toward Effective Digraph Representation Learning: A Magnetic Adaptive Propagation based Approach](https://mlsanity.com/paper/2501.11817/) (2025)
- [Haar-Laplacian for directed graphs](https://mlsanity.com/paper/2411.15527/) (2024)
- [Toward Data-centric Directed Graph Learning: An Entropy-driven Approach](https://mlsanity.com/paper/2505.00983/) (2025)

## Abstract (arXiv)

Recent advancements in Graph Neural Networks have led to state-of-the-art performance on graph representation learning. However, the majority of existing works process directed graphs by symmetrization, which causes loss of directional information. To address this issue, we introduce the magnetic Laplacian, a discrete Schr"odinger operator with magnetic field, which preserves edge directionality by encoding it into a complex phase with an electric charge parameter. By adopting a truncated variant of PageRank named Linear- Rank, we design and build a low-pass filter for homogeneous graphs and a high-pass filter for heterogeneous graphs. In this work, we propose a complex-valued graph convolutional network named Magnetic Graph Convolutional network (MGC). With the corresponding complex-valued techniques, we ensure our model will be degenerated into real-valued when the charge parameter is in specific values. We test our model on several graph datasets including directed homogeneous and heterogeneous graphs. The experimental results demonstrate that MGC is fast, powerful, and widely applicable.