Exact Topological Compliance: Generating Persistence-Equivalent Graphs

Persistent homology summarizes the topological evolution of a graph as persistence diagrams (PD). Generating graphs whose topology matches a target descriptor is increasingly important, yet existing topology-aware generative models rely on soft regularization, producing graphs whose PDs may only be approximately similar. We initiate a principled investigation of the generation problem: produce graphs that realize a target PD exactly. We propose two approaches guided by the implicit local and global constraints a target PD imposes. The local approach builds degree-based PD-equivalent graphs iteratively, and we prove that every output realizes the target PD and that every degree-based PD-equivalent graph is reachable. However, as shown thereafter, local generation necessarily trades off between sample diversity and reaching infeasible states. We therefore recast generation as a global constraint satisfaction problem with a complete encoding solvable by modern constraint programming solvers. This is applicable to any vertex-based permutation equivariant filtration scheme and is minimal under the degree filtration. Experiments on four standard graph generation benchmarks confirm that both methods realize the target PD exactly. Together, our methods provide the first principled treatment of strict topology-compliant graph generation.