Graph Filtrations Overview

• Graph filtrations are parameterized sequences of graphs or complexes that capture multiscale structure through algebraic, topological, and combinatorial methods.
• They enable applications in persistent homology, machine learning, and invariant theory by leveraging robust, multi-parameter frameworks and stability metrics.
• Their computational frameworks facilitate efficient analysis of complex networks while ensuring invariance under perturbations via interleaving and bottleneck distances.

A graph filtration is a structured, parameterized sequence or family of graphs, graph-derived complexes, algebraic objects, or related modules, indexed by a filtration parameter or multi-parameter, typically reflecting increasing complexity, density, scale, or combinatorial structure. Filtrations lie at the interface of combinatorics, topology, algebra, and their algorithmic applications, underpinning a wide range of research directions from topological data analysis and persistent homology to commutative algebra, representation theory, noncommutative ring theory, and high-dimensional statistics.

1. Algebraic and Combinatorial Filtrations on Graphs

Algebraically, filtrations emerge when constructing ideal-theoretic or module-theoretic invariants from graphs. The Koszul filtration for standard graded $K$-algebras, for example, can be constructed for quotient rings defined by binomial edge ideals of “closed” graphs, linking Gröbner-theoretic properties, such as the existence of a quadratic Gröbner basis with respect to reverse lexicographic order, to linear resolutions and “linear flags” in the algebra (inclusion chains of ideals generated by linear forms whose successive quotients are cyclic) (Ene et al., 2013).

A fundamental instance is the filtration by symbolic powers of an ideal. The $v$-number and local $v$-numbers associated to symbolic power filtrations of cover ideals of graphs are asymptotically (quasi-)linear and can be explicitly computed for classes such as complete bipartite graphs, cycles, and complete graphs. The gap between Castelnuovo–Mumford regularity and $v$-number can be made arbitrarily large within both bipartite and non-bipartite graph classes, and the $v$-number is not universally bounded by combinatorial invariants such as the big height (A et al., 14 Mar 2024).

Filtered algebraic $K$-theory is developed by equipping Leavitt path algebras (and corresponding graph $C^*$-algebras) with a filtration via the sublattice of graded (or gauge-invariant) ideals, yielding a structure that enables finer comparison of graph algebras and supports strong classification theorems like the Abrams–Tomforde conjecture (Eilers et al., 2016).

2. Filtrations in Topological and Persistent Homology Contexts

Filtrations are central in topological data analysis (TDA), where a graph (or network) is filtered by a monotonic function, such as edge weights, discrete curvature, or externally assigned scalar fields, to yield a sequence of nested subgraphs, clique complexes, or other cell complexes. This produces a persistence module from which persistent homology, barcodes, and landscapes are extracted.

Filtration functions may be graph-theoretically motivated, such as discrete curvature (Forman–Ricci, Ollivier–Ricci, resistance curvature) (Southern et al., 2023), directionality measures on tournaments (Govc et al., 2020), edge or face birth times in multiparameter bifiltrations (Alonso et al., 2022, Blaser et al., 18 Mar 2025), or cohomological depth in algebraic/topological settings (Felder, 2017). These filtrations capture the multiscale evolution of cycles, connectivity, and higher-order features. The monoidal Rips filtration (Blaser et al., 18 Mar 2025) further generalizes classical constructions by replacing the maximum operator with an arbitrary monoidal product valued in a symmetric monoidal lattice, thereby supporting stable, expressive filtration of arbitrary (possibly directed, weighted, or multiparameter) graphs.

Persistent barcodes, landscapes, and filtration-based kernel coordinates have been shown to be highly expressive and stable under perturbations to the underlying graph or signal function, providing robust graph summarizations that exceed the discriminative power of traditional feature counts or even the 1–Weisfeiler–Lehman (WL) test in certain settings (Southern et al., 2023, Schulz et al., 2021, Zhang et al., 4 Jun 2024).

3. Filtrations in Graph Complexes and Higher Representation Theory

Graph filtrations underpin the structure and paper of advanced algebraic and topological objects. In Kontsevich’s graph complex (as well as internally connected graph complexes), filtrations—such as the depth filtration—confers a grading that corresponds to algebraic or combinatorial depth in Lie algebras (e.g., $\mathfrak{grt}_1$) and supports isomorphisms reflecting the deep combinatorial structure of relations among conjectural generators (Felder, 2017).

Filtered sheaves on moment graphs, constructed over posets with partial orders (for example, quotients of affine Weyl group moment graphs), encode representation-theoretic and geometric invariants, such as indecomposable projective objects in categories arising from modular representation theory and support Verma flag multiplicities or intersection cohomology calculations (Fiebig et al., 2015).

Filtered matchings on the face poset of a simplicial complex or a graph interpolate between discrete Morse complexes and matching complexes, grading the chain complex by the number of cycles induced by a matching; the decomposition of the horizontal (associated graded) homology is controlled by the count and combinatorics of vertex-disjoint cycles or 2-factors (Celoria et al., 2020).

4. Computational, Multiparameter, and Algorithmic Aspects

Computational advances have enabled efficient processing of high-dimensional or multiparameter filtrations. In bifiltered graphs, filtration–domination provides rigorous graph-theoretic criteria for edge removal while preserving essential topological invariants in the corresponding clique complexes. This edge-reduction, achieved using efficient algorithms that operate on the native graph (without explicit complex construction), can eliminate over 90% of edges and dramatically reduce memory and computational costs in multiparameter persistence pipelines (Alonso et al., 2022).

The monoidal Rips filtration supports multiparameter settings, where the lattice of filtration parameters is a product of totally ordered sets, and interleaving distances quantify stability under network or data perturbations. This generalized construction preserves stability guarantees and enables combinations of persistence images from different monoidal products for improved empirical performance in both regression and classification tasks (Blaser et al., 18 Mar 2025).

In graph signal processing, the low persistence filter (LPF) introduced via the Basin Hierarchy Tree (BHT) enables topological denoising of signals on graphs by removing features with low persistence, marking an advance in bridging algebraic topology and signal processing (Lier et al., 26 Aug 2024).

5. Filtrations in Graph Machine Learning: Representation and Kernel Methods

Learning pipelines increasingly exploit graph filtrations to construct expressive global descriptors and readouts. In graph neural network architectures, persistent homology computed over a learnable or carefully defined filtration offers a differentiable, topology-informed aggregation method that is sensitive to multiscale connectivity and higher-order cycles (Hofer et al., 2019, Zhang et al., 4 Jun 2024). Extended persistence methods, incorporating the simultaneous tracking of both homological and cohomological features, can represent arbitrarily long cycles and capture global information not accessible to finite-receptive field message passing networks or the WL[1] test (Zhang et al., 4 Jun 2024).

Graph filtration kernels use feature histograms computed at multiple resolution levels of a filtration, with the inclusion of features such as Weisfeiler–Lehman (WL) vertex labels tracked over the filtration strictly increasing expressivity compared to the classical WL test (Schulz et al., 2021). By aggregating persistence-inspired features, these kernels achieve improved predictive performance, especially on structurally subtle datasets.

Partition-wise graph filtering, as realized in CPF, utilizes graph coarsening to form node partitions and applies distinct polynomial filters to partitions, unifying graph-wise and node-wise filtering paradigms. The first stage leverages structure-aware coarsening; the second uses feature-based $k$-means clustering and further filtering, enabling adaptability to both homophilic and heterophilic regions while controlling the parameterization and risk of overfitting (Li et al., 20 May 2025).

Dynamic graphs introduce temporal structure to filtrations, leading to representations such as filtration surfaces, where filtration curves computed per snapshot stack into a surface or tensor suitable for robust, scalable classification. This approach accommodates evolving node sets, edge weights, and online updating, maintaining linear scaling and parameter-efficiency compared to kernelized or deep temporal methods (Srambical et al., 2023).

6. Special Filtration Frameworks and Examples

• Filtration by Independence Sets and Forests: The forest filtration of a graph successively augments the independence complex until all acyclic vertex subsets are included. Properties such as cohomology dimensions along the filtration relate to combinatorial invariants such as the decycling number and derive upper bounds on related quantities (e.g., Fibonacci numbers of ternary graphs) (Bravo, 2023).
• Filtration in Ergodic/Metric Theory: Basic filtrations in Lebesgue spaces, typically as sequences of measurable partitions with added order and uniformity, admit a classification via combinatorial schemes—measures on the space of hierarchies in $\mathbb Z$—which serve as complete invariants for metric isomorphism in combinatorially definite cases. The universal adic graph construction allows every automorphism to be realized in this combinatorial/filtration-theoretic framework, blending ergodic, combinatorial, and graph-theoretic viewpoints (Vershik et al., 2018).

7. Impact, Expressivity, and Stability

Filtrations, in their various forms—algebraic, topological, combinatorial, or analytic—encode the multiscale or hierarchical structure of graphs and networks, attractor basins, combinatorial invariants, or group actions. Their stability properties—quantified via interleaving distances or bottleneck distances—ensure that persistent invariants are robust under perturbations in the data or structural noise. Moreover, the increased expressivity conferred by tracking features’ “lifespans” (instead of raw counts) enables finer distinctions: for instance, distinguishing strongly regular graphs undetected by message passing GNNs, or distinguishing classes of synthetic graphs that differ only in global cycle structure, tasks at which classical invariant-based or local aggregation methods provably fail.

The unification of graph-wise, node-wise, and partition-wise filtering (as in CPF (Li et al., 20 May 2025)) offers a principled approach to tuning expressivity versus overfitting. Meanwhile, the development of efficient algorithms supporting multi-parameter and large-scale filtrations (e.g., via coarsening, domination criteria, parallel persistence computation) allows their deployment in real-world statistical, algebraic, and geometric machine learning.

In summary, graph filtrations provide a unifying and highly expressive framework linking topology, algebra, combinatorics, and computation, with deep implications for theory, algorithms, and applications spanning from invariant theory to large-scale data analysis and graph-based learning.