Exact Topological Compliance: Generating Persistence-Equivalent Graphs

Persistent homology captures the topological evolution of a graph as a persistence diagram. Creating graphs that satisfy a target topological descriptor is increasingly important, though current generative models typically yield only approximate matches. This paper investigates the problem of generating graphs that exactly realize a target persistence diagram.

Degree-Based Iterative Construction of PD-Equivalent Graphs

The local approach builds degree-based PD-equivalent graphs iteratively. At each step, a new vertex is added together with edges chosen so that the partial persistence diagram of the growing graph remains compatible with the target. The construction respects degree constraints imposed by the target diagram, and the method proves that every output realizes the target PD. Moreover, every degree-based PD-equivalent graph is reachable through this process. However, local generation necessarily faces a compromise between sample diversity and reaching infeasible states when degree constraints conflict with the goal of varying graph structure.

Global Constraint Satisfaction for Topological Exactness

The second recasts generation as a global constraint satisfaction problem. The target persistence diagram imposes implicit constraints on the entire graph structure, and these can be encoded as a complete constraint satisfaction problem for any vertex-based permutation equivariant filtration scheme that is minimal under the degree filtration. Modern constraint programming solvers then search the space of adjacency matrices to find graphs satisfying all constraints simultaneously. This approach does not suffer the diversity limitations of the local method, because the global encoding simultaneously considers all vertices and edges. Both the local and global methods realize the target PD exactly, as confirmed by experiments.

Experimental Evaluation on Graph Generation Benchmarks

The authors report experiments on four standard graph generation benchmarks. Both the local iterative method and the global constraint satisfaction method produce graphs whose persistence diagrams match the target PDs exactly. The results confirm that strict topology-compliant generation is achievable through either approach.

Limitations and Future Directions

The local approach's diversity compromise means that certain graph families may be underrepresented if degree constraints are too restrictive. The global method, while avoiding the diversity issue, relies on the efficiency of the constraint solver and may face computational challenges as graph size increases. Extending the framework to filtration schemes beyond the degree filtration would require additional theoretical development.

Practical Implications for Working Developers

Developers working with topology--aware generative models can adopt the global constraint satisfaction encoding when exact PD matching is required and computational resources permit. The local method offers a faster, iterative alternative when diversity is less critical or when graph size keeps the constraint formulation tractable. Code released with the paper provides a starting point for integration into existing generative pipelines.