Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g.

Feature selection (FS) plays an important role in machine learning, which extracts important features and accelerates the learning process. In this paper, we propose a deep FS method that simultaneously conducts feature selection and differentiable $ k $-NN graph learning based on the Dirichlet Energy. The Dirichlet Energy identifies important features by measuring their smoothness on the graph structure, and facilitates the learning of a new graph that reflects the inherent structure in new feature subspace. We employ Optimal Transport theory to address the non-differentiability issue of learning $ k $-NN graphs in neural networks, which theoretically makes our method applicable to other graph neural networks for dynamic graph learning. Furthermore, the proposed framework is interpretable, since all modules are designed algorithmically.

A fundamental procedure in the analysis of massive datasets is the construction of similarity graphs. Such graphs play a key role for many downstream tasks, including clustering, classification, graph learning, and nearest neighbor search. For these tasks, it is critical to build graphs which are sparse yet still representative of the underlying data. The benefits of sparsity are twofold: firstly, constructing dense graphs is infeasible in practice for large datasets, and secondly, the runtime of downstream tasks is directly influenced by the sparsity of the similarity graph. In this work, we present Stars: a highly scalable method for building extremely sparse graphs via two-hop spanners, which are graphs where similar points are connected by a path of length at most two. Stars can construct two-hop spanners with significantly fewer similarity comparisons, which are a major bottleneck for learning based models where comparisons are expensive to evaluate. Theoretically, we demonstrate that Stars builds a graph in nearly-linear time, where approximate nearest neighbors are contained within two-hop neighborhoods. In practice, we have deployed Stars for multiple data sets allowing for graph building at the Tera-Scale, i.e., for graphs with hundreds of billions of nodes and tens of trillions of edges. We evaluate the performance of Stars for clustering and graph learning, and demonstrate 10~1000-fold improvements in pairwise similarity comparisons and significant running time speedups with negligible quality loss.

Iterative solvers are widely used to accurately simulate physical systems. These solvers require initial guesses to generate a sequence of improving approximate solutions. In this contribution, we introduce a novel method to accelerate iterative solvers for rod dynamics with graph networks (GNs) by predicting the initial guesses to reduce the number of iterations. Unlike existing methods that aim to learn physical systems in an end-to-end manner, our approach guarantees long-term stability and therefore leads to more accurate solutions. Furthermore, our method improves the run time performance of traditional iterative solvers for rod dynamics. To explore our method we make use of position-based dynamics (PBD) as a common solver for physical systems and evaluate it by simulating the dynamics of elastic rods. Our approach is able to generalize across different initial conditions, discretizations, and realistic material properties. We demonstrate that it also performs well when taking discontinuous effects into account such as collisions between individual rods. Finally, to illustrate the scalability of our approach, we simulate complex 3D tree models composed of over a thousand individual branch segments swaying in wind fields.

We thank the reviewers for their valuable comments and for acknowledging the novelty of our work. We hope to address adequately the concerns raised by the reviewers. We apologize for overlooking this. The major computational complexity of our algorithm is the eigenvalue decomposition. ' could be determined by some

Graph Neural Networks are highly effective at learning from relational data, leveraging node and edge features while maintaining the symmetries inherent to graph structures. However, many real-world systems, such as social or biological networks, exhibit complex interactions that are more naturally represented by higher-order topological domains. The emerging field of Geometric and Topological Deep Learning addresses this challenge by introducing methods that utilize and benefit from higher-order structures. Central to TDL is the concept of lifting, which transforms data representations from basic graph forms to more expressive topologies before the application of GNN models for learning. In this work, we propose a structural lifting strategy using Forman-Ricci curvature, which defines an edge-based network characteristic based on Riemannian geometry. Curvature reveals local and global properties of a graph, such as a network's backbones, i.e. coarse, structure-preserving graph geometries that form connections between major communities - most suitably represented as hyperedges to model information flows between clusters across large distances in the network. To this end, our approach provides a remedy to the problem of information distortion in message passing across long distances and graph bottlenecks - a phenomenon known in graph learning as over-squashing.

Graph neural networks (GNNs) are widely used for learning node embeddings in graphs, typically adopting a message-passing scheme. This approach, however, leads to the neighbor explosion problem, with exponentially growing computational and memory demands as layers increase. Graph sampling has become the predominant method for scaling GNNs to large graphs, mitigating but not fully solving the issue. Pre-propagation GNNs (PP-GNNs) represent a new class of models that decouple feature propagation from training through pre-processing, addressing neighbor explosion in theory. Yet, their practical advantages and system-level optimizations remain underexplored. This paper provides a comprehensive characterization of PP-GNNs, comparing them with graph-sampling-based methods in training efficiency, scalability, and accuracy. While PP-GNNs achieve comparable accuracy, we identify data loading as the key bottleneck for training efficiency and input expansion as a major scalability challenge. To address these issues, we propose optimized data loading schemes and tailored training methods that improve PP-GNN training throughput by an average of 15$\times$ over the PP-GNN baselines, with speedup of up to 2 orders of magnitude compared to sampling-based GNNs on large graph benchmarks. Our implementation is publicly available at https://github.com/cornell-zhang/preprop-gnn.

Graph neural networks (GNNs) have achieved significant success in graph representation learning and have been applied to various domains. However, many real-world graphs contain sensitive personal information, such as user profiles in social networks, raising serious privacy concerns when graph learning is performed using GNNs. To address this issue, locally private graph learning protocols have gained considerable attention. These protocols leverage the privacy advantages of local differential privacy (LDP) and the effectiveness of GNN's message-passing in calibrating noisy data, offering strict privacy guarantees for users' local data while maintaining high utility (e.g., node classification accuracy) for graph learning. Despite these advantages, such protocols may be vulnerable to data poisoning attacks, a threat that has not been considered in previous research. Identifying and addressing these threats is crucial for ensuring the robustness and security of privacy-preserving graph learning frameworks. This work introduces the first data poisoning attack targeting locally private graph learning protocols. The attacker injects fake users into the protocol, manipulates these fake users to establish links with genuine users, and sends carefully crafted data to the server, ultimately compromising the utility of private graph learning. The effectiveness of the attack is demonstrated both theoretically and empirically. In addition, several defense strategies have also been explored, but their limited effectiveness highlights the need for more robust defenses.

Recent years have seen a surge in research focused on leveraging graph learning techniques to detect neurodegenerative diseases. However, existing graph-based approaches typically lack the ability to localize and extract the specific brain regions driving neurodegenerative pathology within the full connectome. Additionally, recent works on multimodal brain graph models often suffer from high computational complexity, limiting their practical use in resource-constrained devices. In this study, we present BrainMAP, a novel multimodal graph learning framework designed for precise and computationally efficient identification of brain regions affected by neurodegenerative diseases. First, BrainMAP utilizes an atlas-driven filtering approach guided by the AAL atlas to pinpoint and extract critical brain subgraphs. Unlike recent state-of-the-art methods, which model the entire brain network, BrainMAP achieves more than 50% reduction in computational overhead by concentrating on disease-relevant subgraphs. Second, we employ an advanced multimodal fusion process comprising cross-node attention to align functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) data, coupled with an adaptive gating mechanism to blend and integrate these modalities dynamically. Experimental results demonstrate that BrainMAP outperforms state-of-the-art methods in computational efficiency, without compromising predictive accuracy.