Exact Topological Compliance: Generating Persistence-Equivalent Graphs

Persistent homology summarizes the topological evolution of a graph as persistence diagrams (PDs). 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-e...

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