AI 中文总结
该研究提出关联条码算法，通过监测跨维度代数结构丰富持久同调总结过程，经参考滤过实现稳定性，并在图同构与时序网络关联预测任务中展现判别能力。
AI 中文摘要
著名的持久算法在扫描输入单纯滤过的同时，将同调环的演化总结为所谓的“条码”。我们表明，该总结过程可通过监测贯穿不同维度的其他代数结构来丰富。具体而言，我们提出一种算法，用于监测使p-环成为p-边界并随后演化为(p+1)-环的(p+1)-链，由此得到称为“关联（link）”的额外条，连接持久条码中p维条与(p+1)维条。这些关联产生额外的条码，我们称之为“关联条码（link barcode）”，以区别于标准持久得到的常规条码。关联条码本身不具稳定性，但可通过固定“参考”滤过使其稳定。我们将关联条码应用于图同构问题和时序网络中的关联预测问题，通过这些实验展示其判别能力。
英文摘要
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.
