Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. In this work, we introduce Topological Evolution Rate (TopER), a novel, lowdimensional embedding approach grounded in topological data analysis.

Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem.

Heterogeneity, e.g., due to different types of layers or multiple sub-models, poses

The local inductive bias of message-passing graph neural networks (GNNs) hampers their ability to exploit key structural information (e.g., connectivity and cycles). Positional encoding (PE) and Persistent Homology (PH) have emerged as two promising approaches to mitigate this issue. PE schemes endow GNNs with location-aware features, while PH methods enhance GNNs with multiresolution topological features. However, a rigorous theoretical characterization of the relative merits and shortcomings of PE and PH has remained elusive. We bridge this gap by establishing that neither paradigm is more expressive than the other, providing novel constructions where one approach fails but the other succeeds. Our insights inform the design of a novel learnable method, PiPE (Persistence-informed Positional Encoding), which is provably more expressive than both PH and PE. PiPE demonstrates strong performance across a variety of tasks (e.g., molecule property prediction, graph classification, and out-of-distribution generalization), thereby advancing the frontiers of graph representation learning. Code is available at https://github.com/Aalto-QuML/PIPE.

Graph generative models often face a critical trade-off between learning complex distributions and achieving fast generation speed. We introduce Autoregressive Noisy Filtration Modeling (ANFM), a novel approach that addresses both challenges. ANFM leverages filtration, a concept from topological data analysis, to transform graphs into short sequences of monotonically increasing subgraphs. This formulation extends the sequence families used in previous autoregressive models. To learn from these sequences, we propose a novel autoregressive graph mixer model. Our experiments suggest that exposure bias might represent a substantial hurdle in autoregressive graph generation and we introduce two mitigation strategies to address it: noise augmentation and a reinforcement learning approach. Incorporating these techniques leads to substantial performance gains, making ANFM competitive with state-of-the-art diffusion models across diverse synthetic and real-world datasets. Notably, ANFM produces remarkably short sequences, achieving a 100-fold speedup in generation time compared to diffusion models. This work marks a significant step toward high-throughput graph generation.

Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. In this work, we introduce Topological Evolution Rate (TopER), a novel, low-dimensional embedding approach grounded in topological data analysis. TopER simplifies a key topological approach, Persistent Homology, by calculating the evolution rate of graph substructures, resulting in intuitive and interpretable visualizations of graph data. This approach not only enhances the exploration of graph datasets but also delivers competitive performance in graph clustering and classification tasks. Our TopER-based models achieve or surpass state-of-the-art results across molecular, biological, and social network datasets in tasks such as classification, clustering, and visualization.

Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a comprehensive introduction to two key TML techniques, persistent homology and the Mapper algorithm, with an emphasis on practical applications. Persistent homology captures multi-scale topological features such as clusters, loops, and voids, while the Mapper algorithm creates an interpretable graph summarizing high-dimensional data. To enhance accessibility, we adopt a data-centric approach, enabling readers to gain hands-on experience applying these techniques to relevant tasks. We provide step-by-step explanations, implementations, hands-on examples, and case studies to demonstrate how these tools can be applied to real-world problems. The goal is to equip researchers and practitioners with the knowledge and resources to incorporate TML into their work, revealing insights often hidden from conventional machine learning methods. The tutorial code is available at https://github.com/cakcora/TopologyForML

Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem. Specifically, we establish the necessary and sufficient conditions for distinguishing graphs based on the persistence of their connected components, obtained from filter functions on vertex and edge colors. Our constructions expose the limits of vertex- and edge-level PH, proving that neither category subsumes the other. Leveraging these theoretical insights, we propose RePHINE for learning topological features on graphs. RePHINE efficiently combines vertex- and edge-level PH, achieving a scheme that is provably more powerful than both. Integrating RePHINE into MP-GNNs boosts their expressive power, resulting in gains over standard PH on several benchmarks for graph classification.