A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913113942523904 |
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| author | Bales, Will Rovi, Carmen |
| author_facet | Bales, Will Rovi, Carmen |
| contents | Dynamic Bayesian networks (DBNs) are a widely used framework for modeling systems whose probabilistic structure evolves over time. Standard inference methods focus on local conditional distributions and can miss larger-scale patterns in how dependencies between variables organize and change over time. We introduce a topological approach to this problem. To each DBN we associate a time-varying graph, called a Dynamic Bayesian Graph (DBG), by assigning to each edge a strength that measures variation in its conditional dependence across parent configurations, and retaining edges whose strength exceeds a chosen threshold. We show that this construction fits within the dynamic graph framework of Kim and Mémoli, enabling the use of tools from topological data analysis. Applying persistent homology to a DBG produces a barcode, which records the merging and disappearance of connected groups of strongly dependent variables over time. We prove that this barcode is stable: small perturbations in the conditional probability tables of the DBN lead to small changes in the resulting barcode. This yields a principled and noise-resistant summary of how dependency structure evolves in a dynamic Bayesian network. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11226 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering Bales, Will Rovi, Carmen Algebraic Topology Statistics Theory Machine Learning 55N31 (Primary), 62H22, 62H30 (Secondary) Dynamic Bayesian networks (DBNs) are a widely used framework for modeling systems whose probabilistic structure evolves over time. Standard inference methods focus on local conditional distributions and can miss larger-scale patterns in how dependencies between variables organize and change over time. We introduce a topological approach to this problem. To each DBN we associate a time-varying graph, called a Dynamic Bayesian Graph (DBG), by assigning to each edge a strength that measures variation in its conditional dependence across parent configurations, and retaining edges whose strength exceeds a chosen threshold. We show that this construction fits within the dynamic graph framework of Kim and Mémoli, enabling the use of tools from topological data analysis. Applying persistent homology to a DBG produces a barcode, which records the merging and disappearance of connected groups of strongly dependent variables over time. We prove that this barcode is stable: small perturbations in the conditional probability tables of the DBN lead to small changes in the resulting barcode. This yields a principled and noise-resistant summary of how dependency structure evolves in a dynamic Bayesian network. |
| title | A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering |
| topic | Algebraic Topology Statistics Theory Machine Learning 55N31 (Primary), 62H22, 62H30 (Secondary) |
| url | https://arxiv.org/abs/2605.11226 |