Detecting Homeomorphic 3-manifolds via Graph Neural Networks

• URL: http://arxiv.org/abs/2409.02126v1
• Date: Sun, 1 Sep 2024 12:58:09 GMT
• Title: Detecting Homeomorphic 3-manifolds via Graph Neural Networks
• Authors: Craig Lawrie, Lorenzo Mansi,
• Abstract summary: We study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.

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