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Algorithm for Interpretable Graph Features via Motivic Persistent Cohomology

Yoshihiro Maruyama

TL;DR

This work introduces the Chromatic Persistence Algorithm (CPA), an event-driven method for obtaining refined, cohomological graph features from weighted graphs via graphic arrangements. CPA leverages the chromatic–to–Poincaré/$E$–polynomial correspondence and a deletion–contraction jump identity to update per-threshold invariants and per-edge changes efficiently, yielding a principled, interpretable descriptor of global graph structure. The authors establish exponential worst-case complexity, fixed-parameter tractability in treewidth, and near-linear performance on common graph families, with experimental validation on molecule-like rings showing superior discriminative power over standard 1-skeleton persistent homology. CPA thus provides deterministic, scalable, and interpretable features that can be readily incorporated into graph-ML workflows, complementing local representations and enabling robust analysis of ring/cycle structures in graphs.

Abstract

We present the Chromatic Persistence Algorithm (CPA), an event-driven method for computing persistent cohomological features of weighted graphs via graphic arrangements, a classical object in computational geometry. We establish rigorous complexity results: CPA is exponential in the worst case, fixed-parameter tractable in treewidth, and nearly linear for common graph families such as trees, cycles, and series-parallel graphs. Finally, we demonstrate its practical applicability through a controlled experiment on molecular-like graph structures.

Algorithm for Interpretable Graph Features via Motivic Persistent Cohomology

TL;DR

This work introduces the Chromatic Persistence Algorithm (CPA), an event-driven method for obtaining refined, cohomological graph features from weighted graphs via graphic arrangements. CPA leverages the chromatic–to–Poincaré/–polynomial correspondence and a deletion–contraction jump identity to update per-threshold invariants and per-edge changes efficiently, yielding a principled, interpretable descriptor of global graph structure. The authors establish exponential worst-case complexity, fixed-parameter tractability in treewidth, and near-linear performance on common graph families, with experimental validation on molecule-like rings showing superior discriminative power over standard 1-skeleton persistent homology. CPA thus provides deterministic, scalable, and interpretable features that can be readily incorporated into graph-ML workflows, complementing local representations and enabling robust analysis of ring/cycle structures in graphs.

Abstract

We present the Chromatic Persistence Algorithm (CPA), an event-driven method for computing persistent cohomological features of weighted graphs via graphic arrangements, a classical object in computational geometry. We establish rigorous complexity results: CPA is exponential in the worst case, fixed-parameter tractable in treewidth, and nearly linear for common graph families such as trees, cycles, and series-parallel graphs. Finally, we demonstrate its practical applicability through a controlled experiment on molecular-like graph structures.
Paper Structure (12 sections, 6 theorems, 16 equations, 1 table, 1 algorithm)

This paper contains 12 sections, 6 theorems, 16 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries and Notation
  3. From Graphic Arrangements to Poincaré Polynomials
  4. Chromatic polynomial $\Rightarrow$ Poincaré/$E$–polynomial
  5. Deletion–contraction jump identity (graphic specialization)
  6. Chromatic Persistence Algorithm (CPA) on Graphs
  7. Complexity Analysis
  8. Experimental Verification
  9. Data and filtration.
  10. Methods.
  11. Results.
  12. Conclusion

Key Result

proposition 1

For every $k\ge 0$, $H^k(M(H);\mathbb{Q})$ is pure Hodge--Tate of type $(k,k)$ (all Hodge numbers vanish unless $p=q=k$). Equivalently, where $n=|V|$ and $b_k(H)=\mathrm{rank}\,H^k(M(H);\mathbb{Q})$.

Theorems & Definitions (12)

  • definition 1: Threshold chain
  • proposition 1: Hodge--Tate purity for graphic complements
  • definition 2: Chromatic polynomial
  • definition 3: Poincaré polynomial
  • proposition 2: Chromatic $\Rightarrow$ Poincaré/E; graphic case
  • theorem 1: Deletion–contraction jump identity
  • lemma 1: Component multiplicativity
  • proof
  • theorem 2: Correctness
  • proof
  • ...and 2 more