Graph Multivector Persistence: A Unified Framework for Dynamic Systems

We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a Morse decomposition given by its strongly connected components. As the threshold varies, these multivector fields form a monotone refinement family. We define the Morse persistence diagram by recording the birth and death of Morse sets along this filtration. The construction is purely combinatorial and does not rely on simplicial homology or persistence modules. We prove that the resulting persistence diagram is stable with respect to perturbations of the relation matrix in the sup norm. Each Morse set furthermore carries a combinatorial Conley index, yielding a topologically enriched invariant for multiscale graph structure.