Weisfeiler and Lehman Go Cellular: CW Networks + Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes

1. Van Thuy Hoang
Network Science Lab
Dept. of Artificial Intelligence
The Catholic University of Korea
E-mail: hoangvanthuy90@gmail.com
2024-04-08

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BACKGROUND: Graph Convolutional Networks (GCNs)
• Generate node embeddings based on local network neighborhoods
• Nodes have embeddings at each layer, repeating combine messages
from their neighbor using neural networks

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BACKGROUND
• Two graphs are considered isomorphic if there is a mapping between the nodes of the graphs that preserves
node adjacencies.
• That is, a pair of nodes may be connected by an edge in the first graph if and only if the corresponding pair of
nodes in the second graph is also connected by an edge in the same way.
Graph Isomorphism

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Limitations of GNNs
• Higher-order structures
• Expressive power
• Long range interactions
Weisfeiler and Lehman Go Cellular: CW Networks

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The CWL colouring procedure
• All cells have been assigned unique colours to aid the visualisation of the adjacencies.
Weisfeiler and Lehman Go Cellular: CW Networks

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BACKGROUND
• Molecules as cell complexes
Cellular lifting maps
Weisfeiler and Lehman Go Cellular: CW Networks

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Molecular Message Passing with CW Networks
• The cells in our CW Network receive two types of messages:
• The first specifies messages from atoms to bonds and from bonds
to rings.
• The second type of message, specifies messages between atoms
connected by a bond and messages between bonds that are part of
the same ring
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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Main Contributions
• A novel graph isomorphism test PWL and topological message-passing scheme PCN
operating on path complexes, which encapsulate several theoretical properties of SWL
and CWL
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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Path Complex
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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Path Complex
• S_2-spaces associated with a path complex extended from the graph shown on the left.
• A_2 is a subset of R_2 due to the absence of self-loops in the graph.
For instance
consider 2-path
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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BACKGROUND
• The main difference between our lifting transformation and the ones
• (a) Original graph;
• (b) Simplicial complex, which contains a 2-simplex, 4 1-simplices, and 4 0-simplices,
arising from the original graph.
• (c) Simple path spaces S_2 and S_3 corresponding to the path complex arising from the
original graph.
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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BACKGROUND
• Boundary
• Co-boundary
• Upper-adjacent neighborhood
• Lower-adjacent neighborhood
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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Path Weisfeiler-Lehman Test
• Examples of path complexes arising from (a) a simple path
with length of 3 and (b) a ring with size of 4.
• For every elementary path σ:
Idea of extending graphs to path complexes.
Weisfeiler and Lehman Go Paths: Learning Topological
Features via Path Complexes

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Experiments
• TUDataset Benchmarks
• ZINC
• OGBG-MOLHIV
• Strongly Regular Graphs
Datasets

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Experimental results
• TUDataset Benchmarks
Datasets

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Experimental results
• ZINC & OGBG-MOLHIV Benchmarks
Datasets

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Limitations & Future Works
• High time and space complexities:
• all paths with length k for a graph with n nodes has worst time complexity of O(n
k+1).
• Over-squashing may occur for high-dimensional path complexes
• Increasing the number of message-passing layers can lead to over-smoothing
• An in-depth study into these problems could unlock advancements in higher-order
graph augmentation techniques such as dropout or graph rewiring

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SUMMARY
• How path complexes can be an alternative topological domain for simplicial and regular
cell complexes.
• The versatility of our approach: Universality of paths as the foundational elements of
any graph.