Andreas Roth

Andreas Roth, Thomas Liebig

Our study reveals new theoretical insights into over-smoothing and feature over-correlation in graph neural networks. Specifically, we demonstrate that with increased depth, node representations become dominated by a low-dimensional subspace that depends on the aggregation function but not on the feature transformations. For all aggregation functions, the rank of the node representations collapses, resulting in over-smoothing for particular aggregation functions. Our study emphasizes the importance for future research to focus on rank collapse rather than over-smoothing. Guided by our theory, we propose a sum of Kronecker products as a beneficial property that provably prevents over-smoothing, over-correlation, and rank collapse. We empirically demonstrate the shortcomings of existing models in fitting target functions of node classification tasks.

Andreas Roth, Thomas Liebig

Popular graph neural networks are shallow models, despite the success of very deep architectures in other application domains of deep learning. This reduces the modeling capacity and leaves models unable to capture long-range relationships. The primary reason for the shallow design results from over-smoothing, which leads node states to become more similar with increased depth. We build on the close connection between GNNs and PageRank, for which personalized PageRank introduces the consideration of a personalization vector. Adopting this idea, we propose the Personalized PageRank Graph Neural Network (PPRGNN), which extends the graph convolutional network to an infinite-depth model that has a chance to reset the neighbor aggregation back to the initial state in each iteration. We introduce a nicely interpretable tweak to the chance of resetting and prove the convergence of our approach to a unique solution without placing any constraints, even when taking infinitely many neighbor aggregations. As in personalized PageRank, our result does not suffer from over-smoothing. While doing so, time complexity remains linear while we keep memory complexity constant, independently of the depth of the network, making it scale well to large graphs. We empirically show the effectiveness of our approach for various node and graph classification tasks. PPRGNN outperforms comparable methods in almost all cases.

Andreas Roth, Thomas Liebig

Forecasting future states of sensors is key to solving tasks like weather prediction, route planning, and many others when dealing with networks of sensors. But complete spatial coverage of sensors is generally unavailable and would practically be infeasible due to limitations in budget and other resources during deployment and maintenance. Currently existing approaches using machine learning are limited to the spatial locations where data was observed, causing limitations to downstream tasks. Inspired by the recent surge of Graph Neural Networks for spatio-temporal data processing, we investigate whether these can also forecast the state of locations with no sensors available. For this purpose, we develop a framework, named Forecasting Unobserved Node States (FUNS), that allows forecasting the state at entirely unobserved locations based on spatio-temporal correlations and the graph inductive bias. FUNS serves as a blueprint for optimizing models only on observed data and demonstrates good generalization capabilities for predicting the state at entirely unobserved locations during the testing stage. Our framework can be combined with any spatio-temporal Graph Neural Network, that exploits spatio-temporal correlations with surrounding observed locations by using the network's graph structure. Our employed model builds on a previous model by also allowing us to exploit prior knowledge about locations of interest, e.g. the road type. Our empirical evaluation of both simulated and real-world datasets demonstrates that Graph Neural Networks are well-suited for this task.

Cedric Sanders, Andreas Roth, Thomas Liebig

Over-squashing and over-smoothing are two critical issues, that limit the capabilities of graph neural networks (GNNs). While over-smoothing eliminates the differences between nodes making them indistinguishable, over-squashing refers to the inability of GNNs to propagate information over long distances, as exponentially many node states are squashed into fixed-size representations. Both phenomena share similar causes, as both are largely induced by the graph topology. To mitigate these problems in graph classification tasks, we propose CurvPool, a novel pooling method. CurvPool exploits the notion of curvature of a graph to adaptively identify structures responsible for both over-smoothing and over-squashing. By clustering nodes based on the Balanced Forman curvature, CurvPool constructs a graph with a more suitable structure, allowing deeper models and the combination of distant information. We compare it to other state-of-the-art pooling approaches and establish its competitiveness in terms of classification accuracy, computational complexity, and flexibility. CurvPool outperforms several comparable methods across all considered tasks. The most consistent results are achieved by pooling densely connected clusters using the sum aggregation, as this allows additional information about the size of each pool.

Andreas Roth, Franka Bause, Nils M. Kriege et al.

The ability of message-passing neural networks (MPNNs) to fit complex functions over graphs is limited as most graph convolutions amplify the same signal across all feature channels, a phenomenon known as rank collapse, and over-smoothing as a special case. Most approaches to mitigate over-smoothing extend common message-passing schemes, e.g., the graph convolutional network, by utilizing residual connections, gating mechanisms, normalization, or regularization techniques. Our work contrarily proposes to directly tackle the cause of this issue by modifying the message-passing scheme and exchanging different types of messages using multi-relational graphs. We identify a sufficient condition to ensure linearly independent node representations. As one instantion, we show that operating on multiple directed acyclic graphs always satisfies our condition and propose to obtain these by defining a strict partial ordering of the nodes. We conduct comprehensive experiments that confirm the benefits of operating on multi-relational graphs to achieve more informative node representations.

Andreas Roth, Thomas Liebig

Models with similar performances exhibit significant disagreement in the predictions of individual samples, referred to as prediction churn. Our work explores this phenomenon in graph neural networks by investigating differences between models differing only in their initializations in their utilized features for predictions. We propose a novel metric called Influence Difference (ID) to quantify the variation in reasons used by nodes across models by comparing their influence distribution. Additionally, we consider the differences between nodes with a stable and an unstable prediction, positing that both equally utilize different reasons and thus provide a meaningful gradient signal to closely match two models even when the predictions for nodes are similar. Based on our analysis, we propose to minimize this ID in Knowledge Distillation, a domain where a new model should closely match an established one. As an efficient approximation, we introduce DropDistillation (DD) that matches the output for a graph perturbed by edge deletions. Our empirical evaluation of six benchmark datasets for node classification validates the differences in utilized features. DD outperforms previous methods regarding prediction stability and overall performance in all considered Knowledge Distillation experiments.

Andreas Roth

Graph convolutions have gained popularity due to their ability to efficiently operate on data with an irregular geometric structure. However, graph convolutions cause over-smoothing, which refers to representations becoming more similar with increased depth. However, many different definitions and intuitions currently coexist, leading to research efforts focusing on incompatible directions. This paper attempts to align these directions by showing that over-smoothing is merely a special case of power iteration. This greatly simplifies the existing theory on over-smoothing, making it more accessible. Based on the theory, we provide a novel comprehensive definition of rank collapse as a generalized form of over-smoothing and introduce the rank-one distance as a corresponding metric. Our empirical evaluation of 14 commonly used methods shows that more models than were previously known suffer from this issue.

Andreas Roth

While message-passing neural networks (MPNNs) have shown promising results, their real-world impact remains limited. Although various limitations have been identified, their theoretical foundations remain poorly understood, leading to fragmented research efforts. In this thesis, we provide an in-depth theoretical analysis and identify several key properties limiting the performance of MPNNs. Building on these findings, we propose several frameworks that address these shortcomings. We identify two properties exhibited by many MPNNs: shared component amplification (SCA), where each message-passing iteration amplifies the same components across all feature channels, and component dominance (CD), where a single component gets increasingly amplified as more message-passing steps are applied. These properties lead to the observable phenomenon of rank collapse of node representations, which generalizes the established over-smoothing phenomenon. By generalizing and decomposing over-smoothing, we enable a deeper understanding of MPNNs, more targeted solutions, and more precise communication within the field. To avoid SCA, we show that utilizing multiple computational graphs or edge relations is necessary. Our multi-relational split (MRS) framework transforms any existing MPNN into one that leverages multiple edge relations. Additionally, we introduce the spectral graph convolution for multiple feature channels (MIMO-GC), which naturally uses multiple computational graphs. A localized variant, LMGC, approximates the MIMO-GC while inheriting its beneficial properties. To address CD, we demonstrate a close connection between MPNNs and the PageRank algorithm. Based on personalized PageRank, we propose a variant of MPNNs that allows for infinitely many message-passing iterations, while preserving initial node features. Collectively, these results deepen the theoretical understanding of MPNNs.

Lukas Faber, Sandro Luck, Damian Pascual et al.

The automatic generation of medleys, i.e., musical pieces formed by different songs concatenated via smooth transitions, is not well studied in the current literature. To facilitate research on this topic, we make available a dataset called Medley2K that consists of 2,000 medleys and 7,712 labeled transitions. Our dataset features a rich variety of song transitions across different music genres. We provide a detailed description of this dataset and validate it by training a state-of-the-art generative model in the task of generating transitions between songs.

Martin Burger, Rene Pinnau, Claudia Totzeck et al.

Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit,i.e., for an increasing number of particles.