Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks (2103.03212v2)

Abstract: The pairwise interaction paradigm of graph machine learning has predominantly governed the modelling of relational systems. However, graphs alone cannot capture the multi-level interactions present in many complex systems and the expressive power of such schemes was proven to be limited. To overcome these limitations, we propose Message Passing Simplicial Networks (MPSNs), a class of models that perform message passing on simplicial complexes (SCs). To theoretically analyse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing non-isomorphic SCs. We relate the power of SWL to the problem of distinguishing non-isomorphic graphs and show that SWL and MPSNs are strictly more powerful than the WL test and not less powerful than the 3-WL test. We deepen the analysis by comparing our model with traditional graph neural networks (GNNs) with ReLU activations in terms of the number of linear regions of the functions they can represent. We empirically support our theoretical claims by showing that MPSNs can distinguish challenging strongly regular graphs for which GNNs fail and, when equipped with orientation equivariant layers, they can improve classification accuracy in oriented SCs compared to a GNN baseline.

• The paper introduces the Simplicial WL test that leverages higher-order topological information to distinguish non-isomorphic graphs beyond standard WL tests.
• It enhances GNN expressivity by embedding message passing schemes that integrate orientation and permutation equivariance to handle complex structures.
• The study demonstrates computational efficiency through linear complexity and validates MPSNs’ superiority on diverse real-world graph datasets.

Message Passing Simplicial Networks: A Comprehensive Overview

The paper "Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks" addresses the fundamental limitations inherent in existing graph machine learning models when dealing with higher-order interactions. Unlike traditional graph approaches that primarily focus on pairwise interactions, this work expands upon the capabilities of Graph Neural Networks (GNNs) by introducing Message Passing Simplicial Networks (MPSNs). MPSNs work with simplicial complexes, which allow the representation of complex systems with multi-level interactions. This method leverages the topological structure through what they term the Simplicial Weisfeiler-Lehman (SWL) test, theoretically enhancing expressivity compared to conventional Weisfeiler-Lehman approaches.

Key Contributions

1. Simplicial Weisfeiler-Lehman (SWL) Test: The paper presents a simplicial version of the Weisfeiler-Lehman graph isomorphism test. Utilizing higher-order interactions captured in simplicial complexes, SWL extends the expressive capacity beyond what traditional graph-based tests can achieve. The SWL proves to be more powerful in distinguishing non-isomorphic graph pairs by leveraging the richer topological information available in simplicial complexes.
2. Enhanced Expressivity of MPSNs: By embedding SWL concepts in the neural architecture, MPSNs are strictly more powerful than the WL test and not less than the 3-WL test. The empirical experiments confirm MPSNs' ability to distinguish strongly regular graphs, which are notoriously challenging for standard GNNs. Additionally, the message passing scheme integrates orientation and permutation equivariance, making it robust for complex structural inputs.
3. Complexity and Optimization: The paper explores the computational complexity of MPSNs, emphasizing that by considering specific local adjacencies (boundary and upper adjacencies), MPSN can achieve linear complexity relative to the complex's size. The authors propose a clique complex preprocessing step which serves as an injective map, thus raising the potential to improve representational power without sacrificing efficiency.
4. Theoretical Insights and Number of Linear Regions: By examining the number of linear regions that these models can represent, a novel perspective on the expressive power of MPSNs versus GNNs and SCNNs becomes apparent. This approach highlights MPSNs’ superior capacity to model complex decision boundaries, thereby enhancing their applicability in problems requiring high expressivity.
5. Experimental Validation Across Varied Datasets: The performance of MPSNs validates its theoretical advantages. The architecture successfully distinguishes various types of graph structures and handles real-world graph datasets effectively, showing comparable or superior performance to state-of-the-art GNNs.

Implications and Future Directions

Practically, MPSNs open new avenues for modeling data in domains requiring intricate relational and structural understanding beyond simple graphs. This is relevant for applications in computational biology, network science, and wherever higher-order topological structures are prevalent.

Theoretically, the notions presented could stimulate further development of even more generalized models capable of dealing with other forms of complex networks, such as hypergraphs and beyond. Exploring alternative algebraic structures inherent in data could unlock new potentials for deep learning on complex systems.

An exciting future direction could be the integration of MPSNs within larger machine learning frameworks, enhancing their capability to process multifaceted data inputs such as those found in sensor networks where data might naturally reside on high-dimension manifolds. Another promising avenue might be the development of principled methods to embed and infer connectivity patterns in large-scale systems dynamically, without explicit definition.

In conclusion, the approach laid out in the paper represents a significant step forward for topological data analysis in machine learning, advancing the field with potential impacts across a variety of scientific and engineering disciplines.