Graph Persistence goes Spectral

Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe – a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than PH and spectral information on graphs alone. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.

💡 Research Summary

The paper addresses a fundamental limitation of message‑passing graph neural networks (GNNs): their expressivity is bounded by the Weisfeiler‑Leman (WL) hierarchy and they cannot directly capture higher‑order topological features such as cycles or global connectivity. Persistent Homology (PH) has emerged as a powerful tool to inject topological information into graph learning, but vanilla PH ignores vertex attributes and edge weights, limiting its discriminative power. Recent work (RePHINE) augments PH with color‑based vertex and edge filtrations, yet still fails to incorporate spectral information that reflects the non‑harmonic structure of a graph.

To overcome these gaps, the authors propose SpectRe (Spectral Persistent Homology), a novel graph descriptor that enriches RePHINE diagrams with the eigenvalue spectra of the graph Laplacian computed on the subgraphs that appear during the filtration. Formally, given a coloring filtration (f_v, f_e) on a finite color set X, each 0‑dimensional persistence tuple (birth b(v), death d(v)) is extended with (α(v), γ(v), ρ(v)), where α(v) is the color value, γ(v) is the minimal incident edge weight, and ρ(v) is the multiset of non‑zero Laplacian eigenvalues of the connected component containing v at its death time. Analogously, each 1‑dimensional tuple (birth of a cycle) is appended with the spectrum ρ(e) of the component at the cycle’s birth. The resulting multi‑set SpectRe(G,f) = SpectRe⁰ ⊔ SpectRe¹ has cardinality |V| + β₁(G), matching the size of RePHINE but carrying additional spectral information.

The paper establishes three core theoretical contributions:

1. Isomorphism invariance – Theorem 3.2 proves that SpectRe is unchanged under graph isomorphisms, guaranteeing it is a valid graph invariant.
2. Expressivity hierarchy – Theorem 3.3 shows SpectRe ≻ RePHINE and SpectRe ≻ Laplacian‑Spectrum (LS), while RePHINE and LS remain incomparable. Counter‑examples (Figure 3) illustrate graphs that RePHINE cannot separate but SpectRe can, and vice‑versa for LS.
3. Stability analysis – Classical bottleneck distance provides global stability for ordinary PH, but does not apply to color‑aware diagrams. The authors define new metrics d_RB (for RePHINE) and d_SpecRB (for SpectRe). Theorem 4.3 proves global stability of RePHINE under d_RB, whereas Theorem 4.4 shows SpectRe is locally stable under d_SpecRB, with an explicit bound on instability (Theorem 4.5) that depends on graph complexity. This local stability is sufficient for practical learning scenarios.

Empirically, the authors evaluate SpectRe on 13 synthetic datasets designed to test expressive power and on several real‑world benchmarks (e.g., molecular property prediction, social network classification). SpectRe‑augmented GNNs consistently outperform baselines that use only PH, RePHINE, or raw Laplacian spectra. Ablation studies confirm that the spectral component ρ contributes the majority of the performance gain, especially on tasks where cycle structure is critical. All code and data are released publicly.

In summary, SpectRe unifies topological persistence, vertex/edge attribute information, and spectral signatures into a single, theoretically grounded descriptor. It strictly dominates prior PH‑based graph descriptors in expressive power, enjoys provable (local) stability, and delivers tangible improvements in downstream graph learning tasks. This work paves the way for richer, more discriminative graph representations that can capture both combinatorial and geometric aspects of complex networks.