arXiv subjects

Michael T. Schaub

Publications and source records attributed to Michael T. Schaub.

At least 19 recordsLinked to original sources

arXiv2026

Don't be Afraid of Cell Complexes! An Introduction from an Applied Perspective

Cell complexes (CCs) are a higher-order network model deeply rooted in algebraic topology that has gained interest in signal processing and network science recently. The processing of signals supported on CCs can be described in terms of easily-accessible algebraic or combinatorial notions. However, the commonly presented definition of CCs is grounded in abstract concepts from topology and remains disconnected from the signal processing methods developed for CCs. In this paper, we aim to bridge this gap by providing a simplified definition of CCs that is accessible to a wider audience and can be used in practical applications. Specifically, we first introduce a simplified notion of abstract regular cell complexes (ARCCs). These ARCCs only rely on notions from algebra and are equivalent to regular cell complexes for most practical applications. Second, using this new definition we provide an accessible introduction to (abstract) cell complexes from a perspective of network science and signal processing, including an introduction to central methods and recent developments. Furthermore, as many practical applications work with CCs of dimension 2 and below, we provide an even simpler definition for this case that significantly simplifies understanding and working with CCs in practice.

eess.SP↗

arXiv2026

A repository for discovery and reuse of higher-order network datasets

Higher-order network datasets are dispersed across publications, institutional archives, and software-specific collections, making them difficult to discover, compare, and reuse. We introduce the Aachen Higher-Order Repository of Networks (AHORN), a curated repository of standardized higher-order network datasets derived from publicly released sources. Each dataset entry links a converted dataset to its source, metadata, citation guidance, conversion code, and version history. The repository supports browsable and machine-readable discovery, revision-specific downloads, format validation, and exports for interoperable reuse. We describe the repository architecture, curation workflow, access tools, and the coverage and limitations of the catalog snapshot analyzed in this article.

cs.DL↗

arXiv2026

Robustness of Reinforcement Learning-Based Congestion Management in Low-Voltage Grids

Increases in photovoltaic generation, charging of electric vehicles and heat-pump demand challenge operating limits in low-voltage distribution grids. This requires curative curtailment methods that can operate under sparse observability, noisy measurements, and imperfect grid models. Unlike prior end-to-end reinforcement-learning approaches for partially observable curtailment, this work decouples congestion detection and control by combining a random-forest violation pre-classifier with an actor-critic controller, and evaluates its robustness to measurement noise and grid-parameter mismatch. The framework is tested on a real low-voltage grid using synthetic future operating scenarios with low observability and controllability. With accurate grid parameters, the controller reduces total violation magnitude by 98.9%, and this performance remains nearly unchanged under the tested measurement-noise settings. Grid-model mismatch proves to be more challenging, but the controller still mitigates most violations under the tested mismatch assumptions.

eess.SY↗

arXiv2026

Approximate Simulation-Based Verification of Compatibility of the Friedkin-Johnsen Model with Binary Observations

We consider a verification problem for opinion dynamics based on binary observations. The opinion dynamics is governed by a Friedkin-Johnsen (FJ) model, where only a sequence of binary outputs is available instead of the agents' continuous opinions. At every time-step we observe a binarized output for each agent depending on whether the opinion exceeds a fixed threshold. The objective is to verify whether an FJ model with a given set of stubbornness parameters and initial opinions can generate the observed binary outputs up to a small error. The FJ model is formulated as a transition system, and an approximate simulation relation of two transition systems is defined in terms of the proximity of their opinion trajectories and output sequences. We then construct a finite set of abstract FJ models by simplifying the influence matrix and discretizing the stubbornness parameters and the initial opinions. It is shown that the abstraction approximately simulates any concrete FJ model with continuous parameters and initial opinions, and is itself approximately simulated by some concrete FJ model. These results ensure that consistency verification can be performed over the finite abstraction. Specifically, by checking whether an abstract model satisfies the observation constraints, we can conclude whether the corresponding family of concrete FJ models is consistent with the binary observations. Finally, numerical experiments are presented to illustrate the proposed verification framework.

eess.SY↗

arXiv2026

Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs

Graph neural networks (GNNs) are commonly divided into message-passing neural networks (MPNNs) and spectral GNNs, reflecting two largely separate research traditions in machine learning and signal processing. While MPNNs have a precise definition, there is no widely accepted criterion for what makes a mapping a spectral GNN. Most existing work restricts spectral GNNs to layered architectures based on linear spectral filters. Under this restriction, we show that spectral and spatial GNNs have largely equivalent expressive power. To promote progress in the field, we propose a precise definition of spectral GNNs based on eigenbasis symmetries, in contrast to the definition of MPNNs via neighborhood permutation symmetries. We further argue that the two perspectives offer complementary strengths. MPNNs provide a natural language for discrete structure and expressivity analysis through tools from logic and graph isomorphism, while the spectral perspective offers principled tools for understanding smoothing, bottlenecks, stability, and community structure. Overall, we argue that progress in graph learning will be accelerated by clarifying the similarities and differences between these perspectives and by moving toward a unified theoretical framework.

cs.LG↗

arXiv2026

Phase locking and multistability in the topological Kuramoto model on cell complexes

Higher-order interactions fundamentally shape collective dynamics in oscillator networks. The topological Kuramoto model captures these effects by extending synchronization models to include interactions between cells of arbitrary dimension within simplicial and cell complexes. We introduce the topological nonlinear Kirchhoff conditions to characterize all phase-locked states of the topological Kuramoto model. These states are organized by winding numbers associated with generalized independent cycles, which quantify how phases wind around these cycles. Using rings, Platonic solids, and regular simplices as illustrative examples, we uncover a universal rule: boundaries must have at least five elements for multistability to arise. We further find that independent winding numbers associated with lower- and higher-dimensional boundaries generate cascades of multistability across dimensions. These results show how the topology and boundary structure of cell complexes influence phase locking and multistability, and provide a general framework for collective dynamics on cell complexes.

nlin.AO↗

arXiv2026

Root-to-Leaf Path Random Walks, Normalized Hodge Laplacians, and Cheeger Inequalities on Simplicial Complexes

We introduce root-to-leaf path random walks on double covers of graded signed graphs and analyze their behavior in a general setting. Viewing simplicial complexes within this framework, we show that these walks induce the natural normalization of the coboundary operator and of the Hodge Laplacians while preserving the basic structural features of combinatorial Hodge theory. We then derive Cheeger inequalities for the upper side of the normalized Hodge spectrum, identify the coherent structures governing these bounds, and combine the up- and down-cases into sharper estimates.

math.CO↗

arXiv2026

Higher order trade-offs in hypergraph community detection

Extending community detection from pairwise networks to hypergraphs introduces fundamental theoretical challenges. Hypergraphs exhibit structural heterogeneity with no direct graph analogue: hyperedges of varying orders can connect nodes across communities in diverse configurations, introducing new trade-offs in defining and detecting community structure. We address these challenges by developing a unified framework for community detection in non-uniform hypergraphs under the Hypergraph Stochastic Block Model. We introduce a general signal-to-noise ratio that enables a quantitative analysis of trade-offs unique to higher-order networks, such as which hypergedges we choose to split across communities and how we choose to split them. Building on this framework, we derive a Bethe Hessian operator for non-uniform hypergraphs that provides efficient spectral clustering with principled model selection. We characterize the resulting spectral detectability threshold and compare it to belief propagation limits, showing the methods coincide for uniform hypergraphs but diverge in non-uniform settings. Synthetic experiments confirm our analytical predictions and reveal systematic biases toward preserving higher-order and balanced-shape hyperedges. Application to empirical data demonstrates the practical relevance of these higher-order detectability trade-offs in real-world systems.

cs.SI↗

arXiv2025

Random Abstract Cell Complexes

We define a model for random (abstract) cell complexes (CCs), similiar to the well-known Erdős-Rényi model for graphs and its extensions for simplicial complexes. To build a random cell complex, we first draw from an Erdős-Rényi graph, and consecutively augment the graph with cells for each dimension with a specified probability. As the number of possible cells increases combinatorially -- e.g., 2-cells can be represented as cycles, or permutations -- we derive an approximate sampling algorithm for this model limited to two-dimensional abstract cell complexes. As a basis for this algorithm, we first introduce a spanning-tree-based method that samples simple cycles and allows the efficient approximation of various properties, most notably the probability of occurence of a given cycle. This approximation is of independent interest as it enables the approximation of a wide variety of cycle-related graph statistics using importance sampling. We use this to approximate the number of cycles of a given length on a graph, allowing us to calculate the sampling probability to arrive at a desired expected number of sampled 2-cells. The probability approximation also trivially leads to a sampling algorithm for $2$-cells with a desired sampling probability. We provide some initial analysis into the properties of random CCs drawn from this model. We further showcase practical applications for our random CCs as null models, and in the context of (random) liftings of graphs to cell complexes. The cycle sampling, cycle count estimation, and combined cell sampling algorithms are available in the package `py-raccoon` on the Python Packaging Index.

cs.DS↗

arXiv2025

Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications

Low-pass graph filters are fundamental for signal processing on graphs and other non-Euclidean domains. However, the computation of such filters for parametric graph families can be prohibitively expensive as computation of the corresponding low-frequency subspaces, requires the repeated solution of an eigenvalue problem. We suggest a novel algorithm of low-pass graph filter interpolation based on Riemannian interpolation in normal coordinates on the Grassmann manifold. We derive an error bound estimate for the subspace interpolation and suggest two possible applications for induced parametric graph families. First, we argue that the temporal evolution of the node features may be translated to the evolving graph topology via a similarity correction to adjust the homophily degree of the network. Second, we suggest a dot product graph family induced by a given static graph which allows to infer improved message passing scheme for node classification facilitated by the filter interpolation.

cs.LG↗

arXiv2025

Efficient Sparsification of Simplicial Complexes via Local Densities of States

Simplicial complexes (SCs) have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world datasets often leads to dense SCs, with many higher-order simplicies, which results in prohibitive computational requirements in terms of time and memory consumption. The sparsification of such complexes is thus of broad interest, i.e., the approximation of an original SC with a sparser surrogate SC (with typically only a log-linear number of simplices) that maintains the spectrum of the original SC as closely as possible. In this work, we develop a novel method for a probabilistic sparsification of SCs that uses so-called local densities of states. Using this local densities of states, we can efficiently approximate so-called generalized effective resistance of each simplex, which is proportional to the required sampling probability for the sparsification of the SC. To avoid degenerate structures in the spectrum of the corresponding Hodge Laplacian operators, we suggest a ``kernel-ignoring'' decomposition to approximate the sampling probability. Additionally, we utilize certain error estimates to characterize the asymptotic algorithmic complexity of the developed method. We demonstrate the performance of our framework on a family of Vietoris--Rips filtered simplicial complexes.

stat.ML↗

arXiv2025

Faster Inference of Cell Complexes from Flows via Matrix Factorization

We consider the following inference problem: Given a set of edge-flow signals observed on a graph, lift the graph to a cell complex, such that the observed edge-flow signals can be represented as a sparse combination of gradient and curl flows on the cell complex. Specifically, we aim to augment the observed graph by a set of 2-cells (polygons encircled by closed, non-intersecting paths), such that the eigenvectors of the Hodge Laplacian of the associated cell complex provide a sparse, interpretable representation of the observed edge flows on the graph. As it has been shown that the general problem is NP-hard in prior work, we here develop a novel matrix-factorization-based heuristic to solve the problem. Using computational experiments, we demonstrate that our new approach is significantly less computationally expensive than prior heuristics, while achieving only marginally worse performance in most settings. In fact, we find that for specifically noisy settings, our new approach outperforms the previous state of the art in both solution quality and computational speed.

cs.SI↗

arXiv2025

PyGenStability: Multiscale community detection with generalized Markov Stability

We present PyGenStability, a general-use Python software package that provides a suite of analysis and visualisation tools for unsupervised multiscale community detection in graphs. PyGenStability finds optimized partitions of a graph at different levels of resolution by maximizing the generalized Markov Stability quality function with the Louvain or Leiden algorithms. The package includes automatic detection of robust graph partitions and allows the flexibility to choose quality functions for weighted undirected, directed and signed graphs, and to include other user-defined quality functions.

cs.SI↗

arXiv2025

TopoBench: A Framework for Benchmarking Topological Deep Learning

This work introduces TopoBench, an open-source library designed to standardize benchmarking and accelerate research in topological deep learning (TDL). TopoBench decomposes TDL into a sequence of independent modules for data generation, loading, transforming and processing, as well as model training, optimization and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBench is its support for transformations and lifting across topological domains. Mapping the topology and features of a graph to higher-order topological domains, such as simplicial and cell complexes, enables richer data representations and more fine-grained analyses. The applicability of TopoBench is demonstrated by benchmarking several TDL architectures across diverse tasks and datasets.

cs.LG↗

arXiv2025

Global Ground Metric Learning with Applications to scRNA data

Optimal transport provides a robust framework for comparing probability distributions. Its effectiveness is significantly influenced by the choice of the underlying ground metric. Traditionally, the ground metric has either been (i) predefined, e.g., as the Euclidean distance, or (ii) learned in a supervised way, by utilizing labeled data to learn a suitable ground metric for enhanced task-specific performance. Yet, predefined metrics typically cannot account for the inherent structure and varying importance of different features in the data, and existing supervised approaches to ground metric learning often do not generalize across multiple classes or are restricted to distributions with shared supports. To address these limitations, we propose a novel approach for learning metrics for arbitrary distributions over a shared metric space. Our method provides a distance between individual points like a global metric, but requires only class labels on a distribution-level for training. The learned global ground metric enables more accurate optimal transport distances, leading to improved performance in embedding, clustering and classification tasks. We demonstrate the effectiveness and interpretability of our approach using patient-level scRNA-seq data spanning multiple diseases.

cs.LG↗

arXiv2025

Improving the Noise Estimation of Latent Neural Stochastic Differential Equations

Latent neural stochastic differential equations (SDEs) have recently emerged as a promising approach for learning generative models from stochastic time series data. However, they systematically underestimate the noise level inherent in such data, limiting their ability to capture stochastic dynamics accurately. We investigate this underestimation in detail and propose a straightforward solution: by including an explicit additional noise regularization in the loss function, we are able to learn a model that accurately captures the diffusion component of the data. We demonstrate our results on a conceptual model system that highlights the improved latent neural SDE's capability to model stochastic bistable dynamics.

cs.LG↗

arXiv2025

HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection

In this paper, we propose HLSAD, a novel method for detecting anomalies in time-evolving simplicial complexes. While traditional graph anomaly detection techniques have been extensively studied, they often fail to capture changes in higher-order interactions that are crucial for identifying complex structural anomalies. These higher-order interactions can arise either directly from the underlying data itself or through graph lifting techniques. Our approach leverages the spectral properties of Hodge Laplacians of simplicial complexes to effectively model multi-way interactions among data points. By incorporating higher-dimensional simplicial structures into our method, our method enhances both detection accuracy and computational efficiency. Through comprehensive experiments on both synthetic and real-world datasets, we demonstrate that our approach outperforms existing graph methods in detecting both events and change points.

cs.LG↗

arXiv2025

Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs

Residual connections and normalization layers have become standard design choices for graph neural networks (GNNs), and were proposed as solutions to the mitigate the oversmoothing problem in GNNs. However, how exactly these methods help alleviate the oversmoothing problem from a theoretical perspective is not well understood. In this work, we provide a formal and precise characterization of (linearized) GNNs with residual connections and normalization layers. We establish that (a) for residual connections, the incorporation of the initial features at each layer can prevent the signal from becoming too smooth, and determines the subspace of possible node representations; (b) batch normalization prevents a complete collapse of the output embedding space to a one-dimensional subspace through the individual rescaling of each column of the feature matrix. This results in the convergence of node representations to the top-$k$ eigenspace of the message-passing operator; (c) moreover, we show that the centering step of a normalization layer -- which can be understood as a projection -- alters the graph signal in message-passing in such a way that relevant information can become harder to extract. We therefore introduce a novel, principled normalization layer called GraphNormv2 in which the centering step is learned such that it does not distort the original graph signal in an undesirable way. Experimental results confirm the effectiveness of our method.

cs.LG↗