Persistence Pairing Graphs: Tracking Changes of Selected Homology Bases

Abstract

Persistent homology summarizes a filtration by recording when homological features appear and disappear, but the resulting barcode does not determine a preferred homology basis or preferred cycle representatives. We study the additional information carried by a basis selected through matrix reduction and introduce the \emph{persistence pairing graph}, a reduction-dependent directed graph that records how selected homology classes are re-expressed when their associated persistence intervals end. After fixing a filtered finite CW complex, a coefficient field, a filtration-compatible cell order, and a reduced factorization of the boundary matrix, the reduction determines selected birth cycles. When an associated persistence interval ends, the image of its selected birth class has unique coordinates in the basis formed by selected classes whose intervals survive beyond the same filtration value, and the nonzero coordinates define the graph edges. These coordinates can also be recovered through the Kronecker pairing with a dual cohomology basis. We prove that the active selected birth cycles form a homology basis at every cellwise filtration stage, characterize the kernel and image of each homological transition, and show that the graph is directed and acyclic, with every edge pointing toward an interval containing the source interval. In dimension zero, under the standard reduction convention, the graph agrees with the elder rule merge tree. We also prove that its edges can change while the persistence pairs and barcode remain fixed. Finally, we define a bounded fused graph discrepancy and separate the contribution controlled by ordinary persistence stability from the reduction-dependent graph structure. The graph is therefore a descriptor relative to a fixed reduction convention, rather than an invariant of the persistence module alone.