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.
Persistent homology (PH) has become a cornerstone of topological data analysis, furnishing a mathematically grounded framework for extracting multiscale topological signatures from data [10,11,35,3]. Its ability to summarize structural information robustly has led to successful applications in fields as diverse as computational geometry, machine learning, biology, and network science. These developments have positioned PH as both a theoretical paradigm and a practical tool for analyzing complex data structures.
Yet the standard PH pipeline captures only the ranks of homology groups along a filtration, and in doing so discards finer algebraic information about the space. In many settings, especially when filtrations come from algebraic or combinatorial constructions (for example, spaces coming from hyperplane arrangements or from constructions on graphs), one can enrich the classical barcodes by keeping track of additional structure that refines the underlying invariants. This paper develops a graph-specialized version of that idea and explores its algorithmic consequences.
We introduce the Chromatic Persistence Algorithm (CPA), an event-driven method for analyzing weighted graphs. CPA processes a graph along a natural threshold filtration-adding edges one by one in weight order-and computes at each step: (i) a polynomial that summarizes the global structure of the graph at that threshold, and (ii) a jump, namely the cohomological change that occurs when a new edge is inserted. The algorithm evaluates only at actual “events” (edge insertions), making it conceptually simple and computationally efficient.
The approach relies on two classical results from graph and arrangement theory. The first links the chromatic polynomial χ H (q) of a graph H to the Poincaré polynomial of the complement of its associated arrangement [25,26,8,1], so that graph colorings directly capture topological invariants. In fact, in the Hodge-Tate case relevant to graphic arrangements, the same identity yields the Hodge-Deligne polynomial E(M (H); u, v) by the direct specialization E(M (H); u, v) = χ H (uv). The second ingredient is a deletion-contraction identity, which describes how these invariants change when a single edge is added or contracted [8]. Together, these yield a computable combinatorial foundation for CPA: every update reduces to computing chromatic polynomials and applying an algebraic correction. We establish the complexity guarantees for the Chromatic Persistence Algorithm: exponential time in the worst case, fixed-parameter tractable (FPT) in treewidth, and near-linear on common graph families such as trees, cycles, and series-parallel graphs [14,24,2,5].
Concretely, the problem setting is as follows. Let G = (V, E, w) be a finite weighted graph with distinct edge weights t 1 < • • • < t m . Write H j := G ≤tj for the threshold subgraph and let e j be the unique edge added at step j. For each H j we consider the graphic arrangement A(H j ) ⊂ C |V | and its complement M (H j ). The goal is to compute the Hodge-Deligne polynomial E(M (H j ); u, v) of M (H j ) at all thresholds, together with the jump contributed by the addition of e j . In the Hodge-Tate situation, this reduces to evaluating the chromatic polynomial χ Hj (uv) for each threshold; algorithmically, we compute only the contracted-minor χ Hj /ej (q) and update E j via the deletion-contraction recurrence E j (u, v) = E j-1 (u, v) -χ Hj /ej (uv).
Our approach to the problem can be summarized as follows. We leverage the two identities above to turn the problem into a discrete, event-driven computation. (i) The chromatic→Poincaré identity (Prop. 2) converts topology of the arrangement complement into combinatorics of colorings, so at each threshold we obtain E(M (H j ); u, v) = χ Hj (uv). (ii) The deletion-contraction jump identity (Thm. 1) expresses the change caused by adding one edge; at the E-polynomial level, E(M (H j-1 ); u, v) -E(M (H j ); u, v) = E(M (H j /e j ); u, v). Algorithmically, we therefore process only events (edge insertions): at step j we compute χ Hj /ej (q) and update E j via the deletion-contraction recurrence E j (u, v) = E j-1 (u, v) -χ Hj /ej (uv), together with the barcode zeta update, a generating function that encodes all jumps compactly. This yields a CPA routine with provable guarantees: exponential in the worst case, fixed-parameter tractable in treewidth, and near-linear on trees, cycles, and series-parallel graphs. Experimental verification with molecule-like graphs is provided as well.
From a graph-algorithms viewpoint, CPA reframes persistence on graphs as a sequence of combinatorial updates on chromatic polynomials, exploiting deletion-contraction and dynamic programming (DP) on tree decompositions. This connects topological summarization directly to the Tutte/chromatic toolbox and to parameterized complexity, enabling principled worst-case analyses alongside practical near-linear behavior on sparse, low-treewidth inpu
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