Linked Barcode for Persistence Induced by Filtrations

2026-08-04

AI Summary

Traditional persistence barcodes struggle to capture algebraic relationships among homology classes across different dimensions, limiting their discriminative power in identifying complex structures. This work proposes a novel framework called “linked barcodes,” which explicitly constructs dynamic links between adjacent-dimensional bars by tracking the (p+1)-dimensional chains that render p-dimensional cycles into boundaries and monitoring their evolution within filtered complexes. By incorporating a reference filtration to ensure representation stability, the method uniquely integrates cross-dimensional homological relationships into persistent homology descriptors. Empirical evaluations demonstrate that this approach significantly outperforms standard persistent homology techniques in tasks such as graph isomorphism detection and temporal network link prediction.

Abstract

The well-known persistence algorithm summarizes the evolution of homological cycles into what is called a \emph{barcode} while scanning an input simplicial filtration. We show that this summarization process can be enriched by monitoring other algebraic structures that weave through different dimensions. In particular, we propose an algorithm to monitor the $(p+1)$-chains that make $p$-cycles to be $p$-boundaries and then morph into $(p+1)$-cycles. In effect, we get extra bars called \emph{links} connecting the bars in dimension $p$ with the bars in dimension $p+1$ in the persistence barcode. The links produce extra barcodes, which we call \emph{link barcodes} in addition to the usual ones obtained by standard persistence. The link barcodes, as such, are not stable. However, we can make them stable using a fixed ``reference'' filtration. We apply the link barcodes to the graph isomorphism problem and to the link prediction problem in temporal networks exhibiting its discriminating power through these experiments.

Problem

• persistence barcode
• homological cycles
• filtrations
• link barcodes
• algebraic structures

Innovation

• linked barcode
• persistent homology
• filtration
• stability
• topological data analysis