Mutual Information Free Topological Generalization Bounds via Stability

Providing generalization guarantees for stochastic optimization algorithms is a major challenge in modern learning theory. Recently, several studies highlighted the impact of the geometry of training trajectories on the generalization error, both theoretically and empirically. Among these works, a series of topological generalization bounds have been proposed, relating the generalization error to notions of topological complexity that stem from topological data analysis (TDA). Despite their empirical success, these bounds rely on intricate information-theoretic (IT) terms that can be bounded in specific cases but remain intractable for practical algorithms (such as ADAM), potentially reducing the relevance of the derived bounds. In this paper, we seek to formulate comprehensive and interpretable topological generalization bounds free of intractable mutual information terms. To this end, we introduce a novel learning theoretic framework that departs from the existing strategies via proof techniques rooted in algorithmic stability. By extending an existing notion of \textit{hypothesis set stability}, to \textit{trajectory stability}, we prove that the generalization error of trajectory-stable algorithms can be upper bounded in terms of (i) TDA quantities describing the complexity of the trajectory of the optimizer in the parameter space, and (ii) the trajectory stability parameter of the algorithm. Through a series of experimental evaluations, we demonstrate that the TDA terms in the bound are of great importance, especially as the number of training samples grows. This ultimately forms an explanation of the empirical success of the topological generalization bounds.

Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions

Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric and topological features in a filtration. Although earlier work established global algebraic stability for these operators, the precise change in a single eigenvalue when one simplex, such as a vertex, edge, or triangle, is added has remained unknown. This is important because downstream tools, including heat-kernel signatures and spectral neural networks, depend directly on these eigenvalues. We close this gap by proving a uniform Lipschitz bound: after inserting one simplex, every up-persistent Laplacian eigenvalue can vary by at most twice the Euclidean norm of that simplex's boundary, independent of filtration scale and complex size. This result delivers the first eigenvalue-level robustness guarantee for spectral topological data analysis. It guarantees that spectral features remain stable under local updates and enables reliable error control in dynamic data settings.

The Shape of Consumer Behavior: A Symbolic and Topological Analysis of Time Series

Understanding temporal patterns in online search behavior is crucial for real-time marketing and trend forecasting. Google Trends offers a rich proxy for public interest, yet the high dimensionality and noise of its time-series data present challenges for effective clustering. This study evaluates three unsupervised clustering approaches, Symbolic Aggregate approXimation (SAX), enhanced SAX (eSAX), and Topological Data Analysis (TDA), applied to 20 Google Trends keywords representing major consumer categories. Our results show that while SAX and eSAX offer fast and interpretable clustering for stable time series, they struggle with volatility and complexity, often producing ambiguous ``catch-all'' clusters. TDA, by contrast, captures global structural features through persistent homology and achieves more balanced and meaningful groupings. We conclude with practical guidance for using symbolic and topological methods in consumer analytics and suggest that hybrid approaches combining both perspectives hold strong potential for future applications.

Network Structures as an Attack Surface: Topology-Based Privacy Leakage in Federated Learning

Federated learning systems increasingly rely on diverse network topologies to address scalability and organizational constraints. While existing privacy research focuses on gradient-based attacks, the privacy implications of network topology knowledge remain critically understudied. We conduct the first comprehensive analysis of topology-based privacy leakage across realistic adversarial knowledge scenarios, demonstrating that adversaries with varying degrees of structural knowledge can infer sensitive data distribution patterns even under strong differential privacy guarantees. Through systematic evaluation of 4,720 attack instances, we analyze six distinct adversarial knowledge scenarios: complete topology knowledge and five partial knowledge configurations reflecting real-world deployment constraints. We propose three complementary attack vectors: communication pattern analysis, parameter magnitude profiling, and structural position correlation, achieving success rates of 84.1%, 65.0%, and 47.2% under complete knowledge conditions. Critically, we find that 80% of realistic partial knowledge scenarios maintain attack effectiveness above security thresholds, with certain partial knowledge configurations achieving performance superior to the baseline complete knowledge scenario. To address these vulnerabilities, we propose and empirically validate structural noise injection as a complementary defense mechanism across 808 configurations, demonstrating up to 51.4% additional attack reduction when properly layered with existing privacy techniques. These results establish that network topology represents a fundamental privacy vulnerability in federated learning systems while providing practical pathways for mitigation through topology-aware defense mechanisms.

Topology-Assisted Spatio-Temporal Pattern Disentangling for Scalable MARL in Large-scale Autonomous Traffic Control

Intelligent Transportation Systems (ITSs) have emerged as a promising solution towards ameliorating urban traffic congestion, with Traffic Signal Control (TSC) identified as a critical component. Although Multi-Agent Reinforcement Learning (MARL) algorithms have shown potential in optimizing TSC through real-time decision-making, their scalability and effectiveness often suffer from large-scale and complex environments. Typically, these limitations primarily stem from a fundamental mismatch between the exponential growth of the state space driven by the environmental heterogeneities and the limited modeling capacity of current solutions. To address these issues, this paper introduces a novel MARL framework that integrates Dynamic Graph Neural Networks (DGNNs) and Topological Data Analysis (TDA), aiming to enhance the expressiveness of environmental representations and improve agent coordination. Furthermore, inspired by the Mixture of Experts (MoE) architecture in Large Language Models (LLMs), a topology-assisted spatial pattern disentangling (TSD)-enhanced MoE is proposed, which leverages topological signatures to decouple graph features for specialized processing, thus improving the model's ability to characterize dynamic and heterogeneous local observations. The TSD module is also integrated into the policy and value networks of the Multi-agent Proximal Policy Optimization (MAPPO) algorithm, further improving decision-making efficiency and robustness. Extensive experiments conducted on real-world traffic scenarios, together with comprehensive theoretical analysis, validate the superior performance of the proposed framework, highlighting the model's scalability and effectiveness in addressing the complexities of large-scale TSC tasks.

Navigating the Black Box: Leveraging LLMs for Effective Text-Level Graph Injection Attacks

Text-attributed graphs (TAGs) integrate textual data with graph structures, providing valuable insights in applications such as social network analysis and recommendation systems. Graph Neural Networks (GNNs) effectively capture both topological structure and textual information in TAGs but are vulnerable to adversarial attacks. Existing graph injection attack (GIA) methods assume that attackers can directly manipulate the embedding layer, producing non-explainable node embeddings. Furthermore, the effectiveness of these attacks often relies on surrogate models with high training costs. Thus, this paper introduces ATAG-LLM, a novel black-box GIA framework tailored for TAGs. Our approach leverages large language models (LLMs) to generate interpretable text-level node attributes directly, ensuring attacks remain feasible in real-world scenarios. We design strategies for LLM prompting that balance exploration and reliability to guide text generation, and propose a similarity assessment method to evaluate attack text effectiveness in disrupting graph homophily. This method efficiently perturbs the target node with minimal training costs in a strict black-box setting, ensuring a text-level graph injection attack for TAGs. Experiments on real-world TAG datasets validate the superior performance of ATAG-LLM compared to state-of-the-art embedding-level and text-level attack methods.

Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing

Topological deep learning (TDL) has emerged as a powerful tool for modeling higher-order interactions in relational data. However, phenomena such as oversquashing in topological message-passing remain understudied and lack theoretical analysis. We propose a unifying axiomatic framework that bridges graph and topological message-passing by viewing simplicial and cellular complexes and their message-passing schemes through the lens of relational structures. This approach extends graph-theoretic results and algorithms to higher-order structures, facilitating the analysis and mitigation of oversquashing in topological message-passing networks. Through theoretical analysis and empirical studies on simplicial networks, we demonstrate the potential of this framework to advance TDL.

Comparing the Effects of Persistence Barcodes Aggregation and Feature Concatenation on Medical Imaging

In medical image analysis, feature engineering plays an important role in the design and performance of machine learning models. Persistent homology (PH), from the field of topological data analysis (TDA), demonstrates robustness and stability to data perturbations and addresses the limitation from traditional feature extraction approaches where a small change in input results in a large change in feature representation. Using PH, we store persistent topological and geometrical features in the form of the persistence barcode whereby large bars represent global topological features and small bars encapsulate geometrical information of the data. When multiple barcodes are computed from 2D or 3D medical images, two approaches can be used to construct the final topological feature vector in each dimension: aggregating persistence barcodes followed by featurization or concatenating topological feature vectors derived from each barcode. In this study, we conduct a comprehensive analysis across diverse medical imaging datasets to compare the effects of the two aforementioned approaches on the performance of classification models. The results of this analysis indicate that feature concatenation preserves detailed topological information from individual barcodes, yields better classification performance and is therefore a preferred approach when conducting similar experiments.

Topological Deep Learning for Speech Data

Topological data analysis (TDA) offers novel mathematical tools for deep learning. Inspired by Carlsson et al., this study designs topology-aware convolutional kernels that significantly improve speech recognition networks. Theoretically, by investigating orthogonal group actions on kernels, we establish a fiber-bundle decomposition of matrix spaces, enabling new filter generation methods. Practically, our proposed Orthogonal Feature (OF) layer achieves superior performance in phoneme recognition, particularly in low-noise scenarios, while demonstrating cross-domain adaptability. This work reveals TDA's potential in neural network optimization, opening new avenues for mathematics-deep learning interdisciplinary studies.

GRAIL: A Benchmark for GRaph ActIve Learning in Dynamic Sensing Environments

Graph-based Active Learning (AL) leverages the structure of graphs to efficiently prioritize label queries, reducing labeling costs and user burden in applications like health monitoring, human behavior analysis, and sensor networks. By identifying strategically positioned nodes, graph AL minimizes data collection demands while maintaining model performance, making it a valuable tool for dynamic environments. Despite its potential, existing graph AL methods are often evaluated on static graph datasets and primarily focus on prediction accuracy, neglecting user-centric considerations such as sampling diversity, query fairness, and adaptability to dynamic settings. To bridge this gap, we introduce GRAIL, a novel benchmarking framework designed to evaluate graph AL strategies in dynamic, real-world environments. GRAIL introduces novel metrics to assess sustained effectiveness, diversity, and user burden, enabling a comprehensive evaluation of AL methods under varying conditions. Extensive experiments on datasets featuring dynamic, real-life human sensor data reveal trade-offs between prediction performance and user burden, highlighting limitations in existing AL strategies. GRAIL demonstrates the importance of balancing node importance, query diversity, and network topology, providing an evaluation mechanism for graph AL solutions in dynamic environments.