Persistent Graph Homology

• Persistent graph homology is an algebraic-topological framework that tracks topological features in graphs via nested filtrations, capturing components, cycles, and voids.
• It employs diverse filtrations—such as edge-weight, function-driven, and spectral embeddings—to build simplicial complexes like clique and Vietoris–Rips complexes.
• Applications include graph classification, anomaly detection, and visualization, with strong stability guarantees under small perturbations.

Persistent graph homology is the paper and application of persistent homology—an algebraic-topological framework for capturing multiscale connectivity and higher-order features—applied to graphs and graph-based data. The theory allows for the detection and quantification of evolving topological structures, such as connected components, cycles, and higher-dimensional holes, as a graph undergoes a filtration. Depending on the context, this filtration may arise from edge weights, filtration functions on vertices or edges, embeddings into metric spaces, or more generalized constructions such as closure spaces, digraphs, or lattices of multigraphs.

1. Mathematical Foundations and Constructions

Let $G = (V, E)$ be an (undirected or directed, possibly weighted or colored) graph. The fundamental objects underlying persistent graph homology are:

• Filtration: A nested family of graphs $G_0 \subseteq G_1 \subseteq \cdots \subseteq G_m$ or, more generally, an increasing family of simplicial complexes built on $V$, indexed by a parameter (often thresholded edge weights, node/edge importance scores, or times).
• Simplicial Complexes: For a given graph, the clique complex $K(G)$ or the Vietoris--Rips complex $\mathrm{VR}_\varepsilon(G)$ is constructed by filling in each clique or set of mutually adjacent vertices as a simplex.
• Homology: For each complex in the filtration, homology groups $H_k(K_i; \mathbb{F})$ (typically over $\mathbb{F}_2$) are computed. The rank $\beta_k$ of these groups, the Betti numbers, count $k$-dimensional topological features (e.g., connected components ($k=0$), cycles ($k=1$), voids ($k=2$)).
• Persistence Modules and Diagrams: The sequence $\{ H_k(K_i) \}_{i=0}^m$ and the induced maps form a persistence module, from which the persistence diagram in dimension $k$, $D_k = \{(b_\alpha, d_\alpha)\}$, records the birth and death of features.

Filtration definitions admit various levels of generality:

• Edge-weight filtrations: Add edges in order of increasing weight or decreasing similarity, recovering single-linkage or minimum-spanning-tree-based 0-dimensional PH (Suh et al., 2017).
• Function-driven filtrations: Vertex or edge functions $f: V \cup E \rightarrow \mathbb{R}$ are extended to simplices via $f(\sigma) = \max_{v \in \sigma} f(v)$ and sublevel sets (Ballester et al., 2023).
• Spectral embeddings: Adjacency spectral embedding of a random dot-product graph into $\mathbb{R}^d$ followed by persistent homology on the point cloud (Solanki et al., 2019).
• Generalized settings: Categorical approaches (e.g., closure spaces (Bubenik et al., 2021), lattices of multigraphs (Boils, 4 Feb 2025), DAG-indexed filtrations (Chambers et al., 2014), steady/ranging persistence (Gazull, 9 Jun 2025), directed flag complexes (Chaplin et al., 7 Nov 2024)) subsume and generalize the classical pipeline.

2. Representative Algorithms and Computational Pipelines

Algorithmic computation of persistent graph homology typically involves the following workflow:

1. Filter function or embedding selection: The choice of filter determines the subgraphs or simplicial complexes arising in the filtration.
2. Construction of complexes: For each filtration threshold, build the relevant complex (e.g., flag/clique, Vietoris–Rips, directed flag).
3. Boundary matrix reduction: Compute boundary matrices for each dimension, reducing them by column operations to identify new cycles and holes at each step.
4. Barcode or persistence diagram extraction: Birth and death information for each feature is translated into intervals $(b,d)$.
5. Vectorization and downstream use: Persistence diagrams are vectorized (e.g., Betti curves, landscapes, images) to enable integration with machine learning models or statistical analysis (Ballester et al., 2023, Buffelli et al., 12 Sep 2024, Immonen et al., 2023).

Complexity is determined primarily by the size and structure of the maximal cliques, the filtration length, and the feature dimension $k$. For 0-dimensional PH on a weighted graph, Kruskal’s MST algorithm is optimal ($O(|E| \log |E|)$) (Suh et al., 2017). For higher dimensions, general reductions are cubic or worse in the number of simplices, though for 1-dimensional PH on sparse graphs, optimizations yield near-linear practical performance.

In the context of generalized settings:

• Spectral embedding plus PH: ASE is $O(n^3)$ (or $O(n^2 d)$), PH on point clouds scales as in standard TDA (Solanki et al., 2019).
• Persistent Laplacians: Computation of sparse Laplacian matrices and their spectra enables recovery of Betti numbers and geometric information (Wang et al., 2019).
• DAG persistence: Tree reductions for single-source/sink subgraphs admit an $O(n^4)$ complexity (Chambers et al., 2014).

3. Expressivity, Stability, and Theoretical Guarantees

Persistent graph homology encodes rich structural information:

• Expressivity: PH strictly subsumes color-refinement (1-WL) and, in higher dimensions, can separate graphs indistinguishable by $k$-WL for modest $k$ (Ballester et al., 2023). The expressivity can be tightly characterized via color-separating sets for 0-dimensional PH, and the combination of vertex and edge filtrations (e.g., RePHINE) further increases distinguishing power (Immonen et al., 2023).
• Stability: PH enjoys strong stability properties: small perturbations in the filtration function ($f$), metric ($d$), or weights ($w$) lead to bounded changes in the persistence diagrams (bottleneck distance). This is formalized in the context of Gromov–Hausdorff or interleaving distances (Bubenik et al., 2021, Gazull, 9 Jun 2025, Chaplin et al., 7 Nov 2024). For instance, the bottleneck distance between barcodes is upper-bounded by the sup-norm of the difference in function values or edge weights.
• Consistency: For models with latent position structures (e.g., random dot-product graphs), the persistent homology of spectral embeddings converges almost surely to the true latent-space persistent homology as the number of nodes increases (Solanki et al., 2019).
• Limitations: For higher-dimensional homology ($k\ge 2$), monotonicity and injectivity results for barcodes under path-representable or cost-dominated distances no longer hold (Heo et al., 7 Jan 2025). Non-convex graph features may produce persistent diagrams lacking bottleneck stability (Gazull, 9 Jun 2025).

4. Extensions and Generalizations

The persistent graph homology framework admits multiple significant extensions:

• Generalized filtrations: Beyond linear (totally ordered) filtrations, persistence modules can be indexed by arbitrary partially ordered sets, such as lattices of multigraphs (Boils, 4 Feb 2025) or directed acyclic graphs (DAG-indexed persistence) (Chambers et al., 2014), accommodating more complex dynamic graph processes.
• Directed graphs and flag complexes: Persistent homology on directed flag complexes captures asymmetrical connectivity. Notions of directed homotopy and new stability theorems, specialized to digraphs and their flag complexes, enable robust computation of persistent invariants even in the directed case (Chaplin et al., 7 Nov 2024).
• Categorical perspectives: Categorical TDA formalizes persistence in the language of functor categories, enabling seamless extension to hypergraphs and other combinatorial structures. Convexity of features characterizes those for which steady and ranging persistence coincide and are stable (Gazull, 9 Jun 2025).
• Spectral invariants: Persistent spectral graph theory utilizes Laplacian spectra computed over filtrations, with 0-eigenvalue multiplicity corresponding to persistent Betti numbers, while nonzero spectra encode geometric or dynamical features (e.g., molecular vibration, residue fluctuations) not captured by PH alone (Wang et al., 2019).

5. Applications in Graph Learning and Data Analysis

Persistent graph homology serves as a central descriptor in numerous modern tasks:

• Graph classification and pooling: Integration of PH into GNN architectures (with pooling layers guided by persistence) improves classification accuracy and yields more interpretable topological invariants in learned representations (Ying et al., 26 Feb 2024, Buffelli et al., 12 Sep 2024).
• Higher-order information: Efficient extraction of topological information from high-order clique structures can be achieved by ‘lifting’ the graph to its clique (or r-clique) graph and applying low-dimensional PH, drastically reducing computational cost while retaining key higher-order invariants (Buffelli et al., 12 Sep 2024).
• Anomaly detection: Persistent homology-guided smoothing of edge or node features can clarify normal vs. anomalous behavior in interaction networks, as demonstrated in edge-GNNs for anomaly detection (Yuan et al., 19 Jan 2024).
• Force-directed layouts and visualization: The persistence barcode (notably in dimension 0, equivalent to the MST structure) can be leveraged to control force-directed graph layouts for visualization purposes, revealing global topological shape in otherwise cluttered networks (Suh et al., 2017).
• Statistical topological testing: Bottleneck (or Wasserstein) distances between persistence diagrams can facilitate hypothesis testing for distinguishing network models (e.g., standard vs. mixed-membership SBMs), with consistency guarantees in the large-sample limit (Solanki et al., 2019).
• Comparative shape analysis: Overlays of multiple filtrations (e.g., double subsampling, DAG persistence) enable better detection of “true” topological features over noise (Chambers et al., 2014), and closure-space-based PH formalizes comparison via generalized Gromov–Hausdorff metrics (Bubenik et al., 2021).

6. Limitations, Challenges, and Open Problems

While persistent graph homology supplies a powerful, stable, and highly expressive toolkit, several mathematical and computational challenges remain:

• Filtration design: The effectiveness of PH critically depends on the choice of filtration. Poor or uninformed filter functions can obscure topological signals.
• Computational complexity: High-dimensional PH (e.g., full clique complexes for large graphs) remains prohibitive except for sparse settings or when leveraging algebraic or statistical sparsification techniques. Specific extensions for multigraph lattices and poset-filtrations retain polynomial but potentially high complexity (Boils, 4 Feb 2025).
• Interpretability: Mapping persistent features (bars, diagrams) back to combinatorial events in the graph (e.g., concrete cycles or cuts) can be nontrivial, especially in higher dimensions.
• Stability boundaries: Certain operations (directed edge subdivision outside DAGs, addition of appendage edges, or use of non-convex features) can break stability guarantees for persistent diagrams (Chaplin et al., 7 Nov 2024, Gazull, 9 Jun 2025).
• Generalization beyond graphs: The categorical approach accommodates hypergraphs, but generalizations to continuous combinatorial structures or dynamical processes remain underdeveloped.
• Expressive incompleteness: While PH is at least as powerful as WL, there exist graph families indistinguishable for fixed $k$ and filter, but which are separated only by suitably chosen higher-order or jointly learned filtrations (Ballester et al., 2023, Immonen et al., 2023).

Unresolved questions include the principled design of learned or network-optimized filtrations, scalability of persistence computations for massive or streaming data, deeper links between persistent invariants and global graph properties (e.g., expansion, genus, mixing), and robust generalization to multilayer, temporal, or attributed graphs.

7. Summary Table: Key Frameworks in Persistent Graph Homology

Framework	Filtration Type	Key Objects/Maps	Application Domain
MST/Single-linkage (Suh et al., 2017)	Edge-weight/Similarity	0-dim PH, MST, barcodes	Layout, clustering
Closure space (Bubenik et al., 2021)	Edge weights/Closure operator	Simplicial/cubical filtrations, GH distance	Metrics/stability
Spectral/ASE (Solanki et al., 2019)	Adjacency spectral embedding	PH on embedded point clouds	Model testing
DAG-indexed (Chambers et al., 2014)	Partial-order filtrations	Functors from DAGs to vector spaces	Model alignment
Multigraph lattice (Boils, 4 Feb 2025)	Lattice of merges	Multicomplex, poset-barcode	Neuroscience
Edge/path distance (Heo et al., 7 Jan 2025)	Path-representable distance	Rips complexes under path-based metric	Metric inference
Directed flag (Chaplin et al., 7 Nov 2024)	Ordered digraph filtration	PH on directed flag complex, homotopy	Network topology
Steady/ranging (Gazull, 9 Jun 2025)	Boolean features on sets	Steady/ranging generators, category theory	Generalization
GNN+Persistence (Ballester et al., 2023, Buffelli et al., 12 Sep 2024)	Learned/structural functions	PH+DeepSet integration, pooling with PH	Graph learning

All these frameworks situate persistent homology as a flexible backbone for multiscale topological analysis on graphs, encompassing a spectrum of constructions from combinatorial to high-level categorical, and underpinning a growing suite of practical algorithms for statistical network analysis, machine learning, and data science.