A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering

TL;DR

This paper introduces a topological data analysis method for dynamic Bayesian networks, using persistent homology to produce stable, noise-resistant summaries of evolving dependency structures.

Contribution

It develops a novel topological approach that associates a time-varying graph to DBNs and proves the stability of the resulting barcode for analyzing dependency evolution.

Findings

Persistent homology barcodes effectively summarize dependency changes.

The method is stable under small perturbations in the DBN.

It provides a noise-resistant tool for dynamic structure analysis.

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.