CliquePH: Higher-Order Information for Graph Neural Networks through Persistent Homology on Clique Graphs (2409.08217v2)

Abstract: Graph neural networks have become the default choice by practitioners for graph learning tasks such as graph classification and node classification. Nevertheless, popular graph neural network models still struggle to capture higher-order information, i.e., information that goes \emph{beyond} pairwise interactions. Recent work has shown that persistent homology, a tool from topological data analysis, can enrich graph neural networks with topological information that they otherwise could not capture. Calculating such features is efficient for dimension 0 (connected components) and dimension 1 (cycles). However, when it comes to higher-order structures, it does not scale well, with a complexity of $O(n^d)$, where $n$ is the number of nodes and $d$ is the order of the structures. In this work, we introduce a novel method that extracts information about higher-order structures in the graph while still using the efficient low-dimensional persistent homology algorithm. On standard benchmark datasets, we show that our method can lead to up to $31\%$ improvements in test accuracy.

• The paper introduces CliquePH, a method that integrates higher-order persistent homology information into Graph Neural Networks (GNNs) by applying it efficiently to lifted clique graphs.
• CliquePH computes persistent homology up to dimension one on these higher-order clique graphs, providing a computationally scalable approach that scales linearly with the number of nodes.
• Results show CliquePH achieves significant improvements in test accuracy (up to 31%) on standard benchmarks and enhances the ability of GNNs to distinguish complex graph structures like strongly regular graphs.

Higher-Order Graph Neural Networks via Persistent Homology: CliquePH

This essay provides an expert overview of the paper "CliquePH: Higher-Order Information for Graph Neural Networks Through Persistent Homology on Clique Graphs," which introduces a topological enhancement to Graph Neural Networks (GNNs) aimed at capturing higher-order structures.

Background and Motivation

Graph Neural Networks (GNNs) excel at tasks such as node and graph classification. They predominantly rely on a message-passing framework to model pairwise interactions between nodes. However, modeling higher-order structures like cliques and cycles is challenging due to the limitations inherent in this framework. Traditional approaches often ignore these higher-dimensional interactions, missing significant relational information that could improve model efficacy.

Persistent homology, a tool from topological data analysis, provides a mechanism to capture these higher-order structures through a robust mathematical framework. Nevertheless, persistent homology's computational complexity significantly increases with dimensions, traditionally restricting its use to dimensions up to one. This paper presents CliquePH, a method that integrates higher-order persistent homology information into any GNN, overcoming the computational limitations by using efficient topological techniques.

Methodology

CliquePH introduces a novel topological layer that enhances GNNs by computing persistent homology features. The method involves "lifting" the original graph into a series of clique graphs, representing connections among higher-order structures. Persistent homology is then applied up to dimension one across these lifted graphs. This approach is computationally efficient and scales linearly with the number of nodes due to the use of clique graphs.

Key steps in the methodology include:

1. Graph Lifting: Transforming the original graph into a series of higher order clique graphs.
2. Message Passing: Generating node embeddings using a base GNN.
3. Learnable Persistent Homology: Calculating efficient persistent homology, assigning filtration values to nodes and edges, and embedding persistence diagrams into vector spaces.
4. Information Combination: Integrating persistent homology information with graph node embeddings to enhance representational power.

Results

The paper reports substantial improvements in test accuracy, up to 31%, across several standard benchmark datasets such as MNIST and REDDIT-5K. The effectiveness of CliquePH was benchmarked against existing models like TOGL, demonstrating superior performance and reduced variance in output.

Additionally, CliquePH was tested empirically on the ability to distinguish strongly regular graphs—a challenge for traditional GNNs. The results denote that CliquePH enhances discriminative power beyond that of GNNs enhanced with TOGL, underscoring its efficacy in capturing complex graph structures.

Implications and Future Directions

CliquePH provides both theoretical and practical implications for the field of graph-based learning. It offers a scalable solution to embedding higher-order topological information into neural networks, which may amplify the expressiveness beyond what is achievable with current methods. The method's scalability and flexibility mean it could find application in various domains, from chemistry to social network analysis, where higher-order interactions are pivotal.

From a theoretical standpoint, the integration of persistent homology aligns with growing evidence that these tools can augment the expressive power of GNNs. Notably, CliquePH furthers the expressiveness beyond 1-WL by retaining practicality in computational costs, suggesting that additional research could refine these methods to even higher-dimensional topologies without succumbing to prohibitive complexity.

Future work might explore the integration of more sophisticated homological features or further optimize computation methods for persistent homology within the higher-dimensional space. Moreover, investigating combinations of CliquePH with emergent technologies such as cellular sheaves may yield new insights and potentially lead to even more powerful frameworks in graph-based learning.

Conclusion

In summary, CliquePH represents an important step in the augmentation of GNNs with higher-order topological features, employing efficient computations of persistent homology on clique graphs. The methodology and results presented demonstrate both its scalability and its effect to enhance the performance of neural networks in graph classification, potentially setting a precedent for future advancements in the integration of topological data analysis with machine learning architectures.