Higher-order graph neural networks (HOGNNs) and the related architectures from Topological Deep Learning are an important class of GNN models that harness polyadic relations between vertices beyond plain edges. They have been used to eliminate issues such as over-smoothing or over-squashing, to significantly enhance the accuracy of GNN predictions, to improve the expressiveness of GNN architectures, and for numerous other goals. A plethora of HOGNN models have been introduced, and they come with diverse neural architectures, and even with different notions of what the "higher-order" means. This richness makes it very challenging to appropriately analyze and compare HOGNN models, and to decide in what scenario to use specific ones. To alleviate this, we first design an in-depth taxonomy and a blueprint for HOGNNs. This facilitates designing models that maximize performance. Then, we use our taxonomy to analyze and compare the available HOGNN models. The outcomes of our analysis are synthesized in a set of insights that help to select the most beneficial GNN model in a given scenario, and a comprehensive list of challenges and opportunities for further research into more powerful HOGNNs.

We propose a novel approach to iterated sparse matrix dense matrix multiplication, a fundamental computational kernel in scientific computing and graph neural network training. In cases where matrix sizes exceed the memory of a single compute node, data transfer becomes a bottleneck. An approach based on dense matrix multiplication algorithms leads to suboptimal scalability and fails to exploit the sparsity in the problem. To address these challenges, we propose decomposing the sparse matrix into a small number of highly structured matrices called arrow matrices, which are connected by permutations. Our approach enables communication-avoiding multiplications, achieving a polynomial reduction in communication volume per iteration for matrices corresponding to planar graphs and other minor-excluded families of graphs. Our evaluation demonstrates that our approach outperforms a state-of-the-art method for sparse matrix multiplication on matrices with hundreds of millions of rows, offering near-linear strong and weak scaling.

Graph attention models (A-GNNs), a type of Graph Neural Networks (GNNs), have been shown to be more powerful than simpler convolutional GNNs (C-GNNs). However, A-GNNs are more complex to program and difficult to scale. To address this, we develop a novel mathematical formulation, based on tensors that group all the feature vectors, targeting both training and inference of A-GNNs. The formulation enables straightforward adoption of communication-minimizing routines, it fosters optimizations such as vectorization, and it enables seamless integration with established linear algebra DSLs or libraries such as GraphBLAS. Our implementation uses a data redistribution scheme explicitly developed for sparse-dense tensor operations used heavily in GNNs, and fusing optimizations that further minimize memory usage and communication cost. We ensure theoretical asymptotic reductions in communicated data compared to the established message-passing GNN paradigm. Finally, we provide excellent scalability and speedups of even 4--5x over modern libraries such as Deep Graph Library.

Graph databases (GDBs) enable processing and analysis of unstructured, complex, rich, and usually vast graph datasets. Despite the large significance of GDBs in both academia and industry, little effort has been made into integrating them with the predictive power of graph neural networks (GNNs). In this work, we show how to seamlessly combine nearly any GNN model with the computational capabilities of GDBs. For this, we observe that the majority of these systems are based on, or support, a graph data model called the Labeled Property Graph (LPG), where vertices and edges can have arbitrarily complex sets of labels and properties. We then develop LPG2vec, an encoder that transforms an arbitrary LPG dataset into a representation that can be directly used with a broad class of GNNs, including convolutional, attentional, message-passing, and even higher-order or spectral models. In our evaluation, we show that the rich information represented as LPG labels and properties is properly preserved by LPG2vec, and it increases the accuracy of predictions regardless of the targeted learning task or the used GNN model, by up to 34 compared to graphs with no LPG labels/properties. In general, LPG2vec enables combining predictive power of the most powerful GNNs with the full scope of information encoded in the LPG model, paving the way for neural graph databases, a class of systems where the vast complexity of maintained data will benefit from modern and future graph machine learning methods.

According to the tripartite synapse model, astrocytes have a modulatory effect on neuronal signal transmission. More recently, astrocyte malfunction has been associated with psychiatric diseases such as schizophrenia. Several hypotheses have been proposed on the pathological mechanisms of astrocytes in schizophrenia. For example, post-mortem examinations have revealed a reduced astrocytic density in patients with schizophrenia. Another hypothesis suggests that disease symptoms are linked to an abnormality of glutamate transmission, which is also regulated by astrocytes (glutamate hypothesis of schizophrenia). Electrophysiological findings indicate a dispute over whether the disorder causes an increase or a decrease in neuronal and astrocytic activity. Moreover, there is no consensus as to which molecular pathways and network mechanisms are altered in schizophrenia. Computational models can aid the process in finding the underlying pathological malfunctions. The effect of astrocytes on the activity of neuron-astrocyte networks has been analysed with computational models. These can reproduce experimentally observed phenomena, such as astrocytic modulation of spike and burst signalling in neuron-astrocyte networks. Using an established computational neuron-astrocyte network model, we simulate experimental data of healthy and pathological networks by using different neuronal and astrocytic parameter configurations. In our simulations, the reduction of neuronal or astrocytic cell densities yields decreased glutamate levels and a statistically significant reduction in the network activity. Amplifications of the astrocytic ATP release toward postsynaptic terminals also reduced the network activity and resulted in temporarily increased glutamate levels. In contrast, reducing either the glutamate release or re-uptake in astrocytes resulted in higher network activities. Similarly, an increase in synaptic weights of excitatory or inhibitory neurons raises the excitability of individual cells and elevates the activation level of the network. To conclude, our simulations suggest that the impairment of both neurons and astrocytes disturbs the neuronal network activity in schizophrenia.