Uncovering Hidden Graph Structure for Better Classification

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Author: Denis Avetisyan

A new approach leverages frequent subgraph mining within persistent homology to improve the accuracy of graph classification tasks.

This paper introduces a novel graph filtration method based on frequent subgraph mining to enhance persistent homology for improved graph classification performance.

While persistent homology effectively captures graph topology, existing methods often rely on limited filtrations, overlooking potentially rich structural information. This paper, ‘Frequent subgraph-based persistent homology for graph classification’, introduces a novel filtration technique derived from frequent subgraph mining to generate more informative and stable topological features. Experimental results demonstrate that this approach, integrated into both traditional machine learning models and graph neural networks, achieves state-of-the-art graph classification performance, offering significant gains over benchmark methods. Could this bridge between frequent subgraph mining and topological data analysis unlock even more expressive power for graph representation learning?

Beyond Superficial Features: Unveiling the True Shape of Graphs

Conventional graph analysis frequently demands substantial feature engineering, a process where human experts manually define characteristics to represent nodes and edges. This approach inherently limits the ability to discern a graph’s underlying topological structure – the patterns of connectivity that truly define its organization. Because these engineered features are often tailored to specific tasks, they can fail to generalize to novel scenarios or capture the subtle, yet critical, relationships embedded within the graph itself. Consequently, important information about a graph’s global arrangement, such as the presence of tightly knit communities or influential bridge nodes, can be overlooked, hindering accurate classification or prediction. The reliance on hand-crafted features thus creates a bottleneck, preventing the full potential of graph data from being realized and necessitating more sophisticated methods capable of directly learning from inherent topological properties.

Despite their successes in various graph-based learning tasks, established graph kernels such as Weisfeiler-Lehman and Random Walk Kernel aren’t without limitations. These methods, while capable of discerning structural similarities, often demand significant computational resources, particularly when applied to large and complex networks-the computational cost scales rapidly with network size. Furthermore, these kernels can struggle to capture subtle, high-order topological features; they frequently rely on relatively short-range interactions or simplified representations of the graph’s overall architecture. This means that important distinctions-like the presence of specific motifs, the nuanced arrangement of cycles, or the overall ‘shape’ of the graph-may be overlooked, hindering their ability to effectively classify graphs with complex topologies or identify subtle differences between them.

Advancing classification accuracy hinges on developing methodologies that fully exploit the intrinsic topological properties of graphs. Current approaches frequently rely on hand-crafted features or computationally demanding kernels, potentially overlooking subtle yet critical structural information. A direct integration of topological characteristics-such as connectivity patterns, cycles, and higher-order relationships-promises more robust and efficient classification models. This shift necessitates research into techniques that move beyond feature engineering and instead directly learn from the graph’s inherent shape, offering the potential to unlock superior performance across diverse applications ranging from social network analysis to drug discovery and materials science. Ultimately, leveraging graph topology represents a crucial step towards realizing the full potential of graph-based machine learning.

Persistent Homology: Mapping the Landscape of Graph Structure

Topological Data Analysis (TDA) utilizes techniques from topology to analyze the shape of data, moving beyond traditional metric-based approaches. Persistent homology, a core method within TDA, quantifies topological features – connected components, loops, and voids – present in data at various scales. These features are represented as persistence diagrams, which map the ‘birth’ and ‘death’ of these topological characteristics as a filtration parameter increases. Unlike methods sensitive to noise or minor variations, persistent homology focuses on significant topological features – those persisting over a broad range of scales – providing a robust descriptor of the underlying data shape, irrespective of precise coordinate information. This allows for the comparison of datasets even when expressed in different dimensions or with varying levels of noise, offering insights into data structure beyond simple proximity measures.

Persistent homology analyzes data by constructing a ‘filtration’, a sequence of increasingly complex simplicial complexes. A simplicial complex is a collection of vertices, edges, triangles, and higher-dimensional analogs, built from the data points. The filtration begins with individual vertices and progressively adds higher-dimensional simplices based on a proximity criterion, such as distance. As the filtration progresses, topological features – connected components, loops, voids, and higher-dimensional equivalents – ‘appear’ (are born) and may subsequently ‘disappear’ (die) as the filtration parameter increases. Persistent homology tracks these birth and death times, quantifying the ‘persistence’ of each feature; long-lived features are considered more significant than short-lived ones, allowing for noise reduction and identification of robust structural characteristics within the data.

Representing graphs as topological spaces enables the extraction of features that are invariant to changes in node order and coordinate systems. This is achieved by constructing a simplicial complex from the graph, where nodes become vertices, edges become one-dimensional simplices, and higher-dimensional simplices are formed based on edge connectivity. The resulting topological space focuses on the connectivity of the graph, rather than the specific spatial arrangement or labeling of its nodes. Consequently, isomorphic graphs, or graphs that differ only in node labeling, will yield identical topological representations, ensuring that extracted features are robust to these transformations. This approach allows for the comparison of graph structures irrespective of how they are initially represented or embedded, facilitating analysis across diverse datasets and representations.

The Bottleneck Distance is a metric used to quantify the dissimilarity between two persistence diagrams. It is defined as the maximum distance between any point in one diagram and its nearest neighbor in the other diagram. Formally, given two persistence diagrams D_1 and D_2, the Bottleneck Distance is calculated as dist_{B}(D_1, D_2) = \max_{x \in D_1} \min_{y \in D_2} ||x - y||_{\in fty}, where || \cdot ||_{\in fty} denotes the supremum norm (maximum absolute difference between coordinates). This metric is particularly useful because it focuses on the most significant topological features – those with the largest persistence – and is stable under small perturbations in the input data, making it robust for comparing the topological characteristics of different graphs or datasets.

Frequency-Based Filtration: A Scalable Path to Topological Insight

Frequent subgraph mining is a technique used to discover statistically significant and recurring substructures within a larger graph. Algorithms in this domain, such as those utilizing Apriori-like approaches or pattern growth methods, identify subgraphs that appear with a frequency exceeding a user-defined threshold. These frequent subgraphs are not merely common; their prevalence suggests they represent inherent organizational principles or shared characteristics within the graph’s connectivity. The identified subgraphs are then utilized as building blocks for constructing a filtration, where each level of the filtration corresponds to the inclusion of subgraphs up to a specific size or frequency, ultimately providing a structured basis for topological analysis via persistent homology.

Frequent Subgraph Filtration constructs a filtration for persistent homology by ordering simplices based on the frequency of their containing subgraphs. Traditional filtration methods often rely on geometric criteria like distance or density, which can be computationally expensive for large graphs. By utilizing pre-computed frequent subgraphs – those appearing above a defined support threshold – the filtration sequence is built by incrementally adding simplices that correspond to these frequent patterns. This approach reduces computational complexity because the number of frequent subgraphs is typically significantly smaller than the total number of possible simplices. The resulting filtration, while topologically equivalent to other methods, allows for faster computation of persistent homology, particularly in datasets where identifying recurring motifs is a primary goal. This efficiency gain enables the scalable computation of \text{Frequency-based Persistent Homology} features for downstream analysis.

MNI-Frequent Subgraph mining employs a modified neighborhood intersection count to identify statistically significant subgraphs. This technique addresses limitations of standard frequent subgraph mining by focusing on subgraph density within local neighborhoods, thereby reducing the impact of sparse or isolated patterns. Specifically, the MNI count assesses the overlap of neighborhoods containing a given subgraph, providing a more robust measure of subgraph prevalence than simple support counts. This refined approach prioritizes the identification of subgraphs that are not only frequent but also locally representative, leading to more stable and interpretable results in downstream topological data analysis and persistent homology calculations.

Frequency-based Persistent Homology (FPH) offers a scalable alternative to traditional persistent homology calculations by focusing on topological features associated with frequent subgraphs. This method computes persistence diagrams based on the appearance frequency of specific subgraphs within a larger network, reducing computational complexity compared to methods that consider all possible topological carriers. The resulting persistence diagrams, representing birth and death times of topological features, are directly applicable as input features for machine learning algorithms, enabling tasks such as classification, regression, and clustering based on network topology. The scalability of FPH stems from limiting the search space to frequent subgraphs, thereby reducing the number of homology groups that require computation and allowing analysis of significantly larger datasets than conventional approaches.

Topology-Aware Graph Learning: Bridging TDA and Graph Neural Networks

Frequency-based Persistent Homology (FPH) offers a novel approach to graph classification by directly leveraging topological features extracted from network structures. Unlike traditional methods that rely heavily on node attributes or graph statistics, FPH characterizes graphs through the lifespan of their topological features – specifically, connected components and cycles – as revealed by analyzing a frequency-based representation of the graph. This allows the model to discern underlying patterns and global structures that might be missed by local feature analysis. By quantifying these topological characteristics, FPH-ML establishes a powerful, feature-agnostic pathway for distinguishing between different graph classes, offering a robust alternative when node or edge attributes are limited or unreliable. The resulting features provide a concise and informative representation of graph shape, facilitating improved classification performance and offering insights into the intrinsic geometry of complex networks.

The integration of topological data analysis with graph neural networks, realized through ‘FPH-GNN’, represents a significant advancement in graph representation learning. By incorporating features derived from persistent homology – specifically, frequency-based persistent homology – these models move beyond traditional node and edge attributes to capture the underlying shape and connectivity of graphs. This allows ‘FPH-GNN’ to discern subtle structural differences that might be missed by conventional graph neural networks, leading to more robust and informative graph embeddings. The result is an enhanced capacity to learn complex relationships within graph data, improving performance on tasks such as graph classification and node prediction by providing a more complete and nuanced understanding of the graph’s inherent topology.

Recent advancements in graph classification demonstrate a significant performance boost through the incorporation of topological data analysis. These models, leveraging the inherent shape and connectivity of graphs, consistently outperform traditional Graph Convolutional Networks (GCN) and Graph Isomorphism Networks (GIN). Evaluations reveal accuracy gains ranging from 0.4 to 21 percent across diverse datasets, indicating that explicitly representing a graph’s topological features-such as loops and voids-provides crucial information often missed by methods focused solely on node attributes and immediate connections. This improvement is particularly notable on complex datasets like DD, where integration of these features yielded an impressive accuracy of up to 82.94 percent-an 8.2 point increase over baseline GCN/GIN architectures.

Evaluations demonstrate the efficacy of integrating Frequency-based Persistent Homology into graph neural networks, achieving a peak accuracy of 82.94% when tested on the DD dataset. This represents a substantial improvement over standard Graph Convolutional Network (GCN) and Graph Isomorphism Network (GIN) architectures, with the models consistently exceeding their performance by as much as 8.2 percentage points. These gains suggest that incorporating topological features derived from persistent homology allows the networks to capture more nuanced structural information within the graphs, leading to more accurate classifications and a stronger ability to generalize to unseen data.

The presented work emphasizes a holistic approach to graph classification, recognizing that the structure of a graph dictates its behavior. This aligns with the principle that simplicity scales, cleverness does not; by focusing on frequent subgraphs as a basis for filtration in persistent homology, the research avoids overly complex representations. The method’s success demonstrates a preference for understandable, foundational elements over intricate, specialized features. As Barbara Liskov stated, “Programs must be correct and usable. The usability is important because programs are used by people.” This resonates with the need for interpretable topological features – features which, while mathematically rigorous, must ultimately contribute to a usable classification model. The reliance on frequent subgraphs, therefore, isn’t merely a technical choice but an architectural one, prioritizing a scalable, fundamentally sound approach to topological data analysis.

The Road Ahead

The pursuit of robust graph classification invariably returns the researcher to fundamental questions of representation. This work, while demonstrating the efficacy of frequent subgraph mining within a persistent homology framework, merely shifts the locus of complexity. One cannot simply append a more refined filtration to an existing classifier and expect systemic improvement; the entire architecture must accommodate the nuanced information now being presented. To treat topological features as isolated variables is akin to examining a single organ without considering the circulatory system that sustains it.

Future efforts should concentrate on integrating these topological priors directly into the learning process, perhaps through novel graph neural network layers designed to explicitly leverage persistent homology. The current paradigm often feels iterative – refine the topological signal, then attempt to force-fit it into a geometric deep learning structure. A more elegant solution would demand a unified framework where topology and geometry are inseparable from the outset.

Ultimately, the true measure of this approach will not be incremental gains in accuracy, but a demonstrable increase in the interpretability of graph classification models. The ability to discern why a particular graph receives a given label, rooted in its underlying topological structure, remains the most challenging, and arguably the most vital, frontier.

Original article: https://arxiv.org/pdf/2512.24917.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-05 01:27