Linked Barcode for Persistence Induced by Filtrations

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.