A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering
Authors: Will Bales, Carmen Rovi
Summary
The authors introduce a topological approach to analyzing Dynamic Bayesian Networks (DBNs) by converting them into time-varying graphs where edge strength measures variation in conditional dependence across parent configurations. Applying persistent homology to these graphs produces a "barcode" summarizing how groups of strongly dependent variables merge and disappear over time, and they prove this barcode is stable (robust to small perturbations in the conditional probability tables).
Main takeaways:
- Standard DBN inference focuses on local conditional distributions and can miss large-scale patterns in dependency structure.
- They assign each edge a strength measuring conditional dependence variation, retaining strong edges above a threshold.
- Persistent homology produces a barcode recording when connected groups of strongly dependent variables merge or disappear.
- The barcode is stable: small perturbations in the DBN's conditional probability tables lead to small barcode changes.
- Provides a noise-resistant summary of evolving dependency structure in dynamic Bayesian networks.
Relevance
Not directly related to my persona/midtraining work — this is about probabilistic graphical models and topological data analysis, not about language model behavioral installation or persona geometry.
Abstract
arXiv:2605.11226v1 Announce Type: cross Abstract: 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 M'emoli, 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.