Edge Directionality Improves Learning on Heterophilic Graphs (2305.10498v3)

Abstract: Graph Neural Networks (GNNs) have become the de-facto standard tool for modeling relational data. However, while many real-world graphs are directed, the majority of today's GNN models discard this information altogether by simply making the graph undirected. The reasons for this are historical: 1) many early variants of spectral GNNs explicitly required undirected graphs, and 2) the first benchmarks on homophilic graphs did not find significant gain from using direction. In this paper, we show that in heterophilic settings, treating the graph as directed increases the effective homophily of the graph, suggesting a potential gain from the correct use of directionality information. To this end, we introduce Directed Graph Neural Network (Dir-GNN), a novel general framework for deep learning on directed graphs. Dir-GNN can be used to extend any Message Passing Neural Network (MPNN) to account for edge directionality information by performing separate aggregations of the incoming and outgoing edges. We prove that Dir-GNN matches the expressivity of the Directed Weisfeiler-Lehman test, exceeding that of conventional MPNNs. In extensive experiments, we validate that while our framework leaves performance unchanged on homophilic datasets, it leads to large gains over base models such as GCN, GAT and GraphSage on heterophilic benchmarks, outperforming much more complex methods and achieving new state-of-the-art results.

• The paper introduces Dir-GNN, a framework that separately aggregates incoming and outgoing edges to leverage directionality and enhance learning on heterophilic graphs.
• It outperforms traditional GNNs on heterophilic benchmarks while maintaining similar computational complexity.
• The work emphasizes the importance of modeling asymmetry in directed graphs for effective network analysis.

Overview of Dir-GNN

Graph Neural Networks (GNNs) have seen considerable use for analyzing data with relational structure. To make sense of this data, many current GNN models make an underlying assumption that the input graph is undirected, which implies that the information flows in a symmetric and undifferentiated way between connected nodes. However, directed graphs are pervasive, representing a wide range of systems in which the relationships are asymmetric—think of the World Wide Web or a predator-prey network in ecology. In this paper, the researchers shed light on the drawbacks of simplistically transforming directed graphs into undirected ones and introduce Directed Graph Neural Network (Dir-GNN), an innovative framework designed specifically for directed graphs.

Motivation For Directed Models

The common practice of converting directed graphs into undirected ones hinges on historical constraints and the success of early benchmarks on homophilic graphs, where similar nodes are connected. Crucially, ignoring the edge directionality can lose information essential for tasks involving directed networks, which can deteriorate a model’s performance, particularly on heterophilic graphs where dissimilar nodes tend to connect.

The Dir-GNN Framework

The key contribution, Dir-GNN, is a versatile and more expressive framework that leverages edge directionality for enhanced learning on heterophilic graphs. It consists of performing separate aggregations for incoming and outgoing edges and can be applied to any Message Passing Neural Network. The framework's added complexity in capturing directed information does not just enrich the representational capacity—it respects the Directed Weisfeiler-Lehman test and consistently outperforms existing GNN architectures, particularly on heterophilic benchmarks. It achieves this while maintaining computational complexity similar to undirected counterparts, making it a practical extension rather than a completely new model requiring extensive computational resources.

Unveiling The Relationship with Edge Directionality

Through extensive experimentation, the research demonstrates that including edge directionality amplifies heterophilic graphs' effective homophily, translating into significant performance gains. It's worth noting that in homophilic settings, where similarity correlates with connectedness, the incorporation of directionality leaves performance relative unchanged—indicating that this framework's benefits are most profound when facing the learning challenges presented by heterophily.

Conclusion and Future Work

Dir-GNN marks an advance in the graph neural network field, raising awareness of the limitations imposed by the largely unchallenged convention of undirected models and practically demonstrating the untapped potential of directed models. Although its investigation into the expressivity of Dir-GNN within the context of heterophilic graphs has offered positive indications, the avenue for future research is wide open. Future enhancements could explore theoretical aspects of effective homophily in directed graphs and explore different ways to aggregate information from incoming and outgoing edges.

The full code for the research presented in this paper is available at the dedicated GitHub repository.