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Francesco Di Giovanni, T. Konstantin Rusch, Michael M. Bronstein, Andreea Deac, Marc Lackenby, Siddhartha Mishra, Petar Veličković

Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). Given their widespread use, understanding the expressive power of MPNNs is a key question. However, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of pairwise interactions between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that over-squashing hinders the expressive power of MPNNs. We validate our theoretical findings through extensive controlled experiments and ablation studies.

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T. Konstantin Rusch, Michael M. Bronstein, Siddhartha Mishra

Node features of graph neural networks (GNNs) tend to become more similar with the increase of the network depth. This effect is known as over-smoothing, which we axiomatically define as the exponential convergence of suitable similarity measures on the node features. Our definition unifies previous approaches and gives rise to new quantitative measures of over-smoothing. Moreover, we empirically demonstrate this behavior for several over-smoothing measures on different graphs (small-, medium-, and large-scale). We also review several approaches for mitigating over-smoothing and empirically test their effectiveness on real-world graph datasets. Through illustrative examples, we demonstrate that mitigating over-smoothing is a necessary but not sufficient condition for building deep GNNs that are expressive on a wide range of graph learning tasks. Finally, we extend our definition of over-smoothing to the rapidly emerging field of continuous-time GNNs.

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Benjamin Paul Chamberlain, Sergey Shirobokov, Emanuele Rossi, Fabrizio Frasca, Thomas Markovich, Nils Hammerla, Michael M. Bronstein, Max Hansmire

Many Graph Neural Networks (GNNs) perform poorly compared to simple heuristics on Link Prediction (LP) tasks. This is due to limitations in expressive power such as the inability to count triangles (the backbone of most LP heuristics) and because they can not distinguish automorphic nodes (those having identical structural roles). Both expressiveness issues can be alleviated by learning link (rather than node) representations and incorporating structural features such as triangle counts. Since explicit link representations are often prohibitively expensive, recent works resorted to subgraph-based methods, which have achieved state-of-the-art performance for LP, but suffer from poor efficiency due to high levels of redundancy between subgraphs. We analyze the components of subgraph GNN (SGNN) methods for link prediction. Based on our analysis, we propose a novel full-graph GNN called ELPH (Efficient Link Prediction with Hashing) that passes subgraph sketches as messages to approximate the key components of SGNNs without explicit subgraph construction. ELPH is provably more expressive than Message Passing GNNs (MPNNs). It outperforms existing SGNN models on many standard LP benchmarks while being orders of magnitude faster. However, it shares the common GNN limitation that it is only efficient when the dataset fits in GPU memory. Accordingly, we develop a highly scalable model, called BUDDY, which uses feature precomputation to circumvent this limitation without sacrificing predictive performance. Our experiments show that BUDDY also outperforms SGNNs on standard LP benchmarks while being highly scalable and faster than ELPH.

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T. Konstantin Rusch, Benjamin P. Chamberlain, Michael W. Mahoney, Michael M. Bronstein, Siddhartha Mishra

We present Gradient Gating (G$^2$), a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which the multi-rate gradient gating mechanism is built. We rigorously prove that G$^2$ alleviates the oversmoothing problem and allows the design of deep GNNs. Empirical results are presented to demonstrate that the proposed framework achieves state-of-the-art performance on a variety of graph learning tasks, including on large-scale heterophilic graphs.

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Fabrizio Frasca, Beatrice Bevilacqua, Michael M. Bronstein, Haggai Maron

Subgraph GNNs are a recent class of expressive Graph Neural Networks (GNNs) which model graphs as collections of subgraphs. So far, the design space of possible Subgraph GNN architectures as well as their basic theoretical properties are still largely unexplored. In this paper, we study the most prominent form of subgraph methods, which employs node-based subgraph selection policies such as ego-networks or node marking and deletion. We address two central questions: (1) What is the upper-bound of the expressive power of these methods? and (2) What is the family of equivariant message passing layers on these sets of subgraphs?. Our first step in answering these questions is a novel symmetry analysis which shows that modelling the symmetries of node-based subgraph collections requires a significantly smaller symmetry group than the one adopted in previous works. This analysis is then used to establish a link between Subgraph GNNs and Invariant Graph Networks (IGNs). We answer the questions above by first bounding the expressive power of subgraph methods by 3-WL, and then proposing a general family of message-passing layers for subgraph methods that generalises all previous node-based Subgraph GNNs. Finally, we design a novel Subgraph GNN dubbed SUN, which theoretically unifies previous architectures while providing better empirical performance on multiple benchmarks.

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Francesco Di Giovanni, James Rowbottom, Benjamin P. Chamberlain, Thomas Markovich, Michael M. Bronstein

Dynamical systems minimizing an energy are ubiquitous in geometry and physics. We propose a gradient flow framework for GNNs where the equations follow the direction of steepest descent of a learnable energy. This approach allows to explain the GNN evolution from a multi-particle perspective as learning attractive and repulsive forces in feature space via the positive and negative eigenvalues of a symmetric "channel-mixing" matrix. We perform spectral analysis of the solutions and conclude that gradient flow graph convolutional models can induce a dynamics dominated by the graph high frequencies which is desirable for heterophilic datasets. We also describe structural constraints on common GNN architectures allowing to interpret them as gradient flows. We perform thorough ablation studies corroborating our theoretical analysis and show competitive performance of simple and lightweight models on real-world homophilic and heterophilic datasets.

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Emanuele Rossi, Federico Monti, Yan Leng, Michael M. Bronstein, Xiaowen Dong

Strategic interactions between a group of individuals or organisations can be modelled as games played on networks, where a player's payoff depends not only on their actions but also on those of their neighbours. Inferring the network structure from observed game outcomes (equilibrium actions) is an important problem with numerous potential applications in economics and social sciences. Existing methods mostly require the knowledge of the utility function associated with the game, which is often unrealistic to obtain in real-world scenarios. We adopt a transformer-like architecture which correctly accounts for the symmetries of the problem and learns a mapping from the equilibrium actions to the network structure of the game without explicit knowledge of the utility function. We test our method on three different types of network games using both synthetic and real-world data, and demonstrate its effectiveness in network structure inference and superior performance over existing methods.

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Ahmed A. A. Elhag, Gabriele Corso, Hannes Stärk, Michael M. Bronstein

Traditional Graph Neural Networks (GNNs) rely on message passing, which amounts to permutation-invariant local aggregation of neighbour features. Such a process is isotropic and there is no notion of `direction' on the graph. We present a new GNN architecture called Graph Anisotropic Diffusion. Our model alternates between linear diffusion, for which a closed-form solution is available, and local anisotropic filters to obtain efficient multi-hop anisotropic kernels. We test our model on two common molecular property prediction benchmarks (ZINC and QM9) and show its competitive performance.

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Kamilia Mullakaeva, Luca Cosmo, Anees Kazi, Seyed-Ahmad Ahmadi, Nassir Navab, Michael M. Bronstein

Graphs are a powerful tool for representing and analyzing unstructured, non-Euclidean data ubiquitous in the healthcare domain. Two prominent examples are molecule property prediction and brain connectome analysis. Importantly, recent works have shown that considering relationships between input data samples have a positive regularizing effect for the downstream task in healthcare applications. These relationships are naturally modeled by a (possibly unknown) graph structure between input samples. In this work, we propose Graph-in-Graph (GiG), a neural network architecture for protein classification and brain imaging applications that exploits the graph representation of the input data samples and their latent relation. We assume an initially unknown latent-graph structure between graph-valued input data and propose to learn end-to-end a parametric model for message passing within and across input graph samples, along with the latent structure connecting the input graphs. Further, we introduce a degree distribution loss that helps regularize the predicted latent relationships structure. This regularization can significantly improve the downstream task. Moreover, the obtained latent graph can represent patient population models or networks of molecule clusters, providing a level of interpretability and knowledge discovery in the input domain of particular value in healthcare.

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Cristian Bodnar, Francesco Di Giovanni, Benjamin Paul Chamberlain, Pietro Liò, Michael M. Bronstein

Cellular sheaves equip graphs with "geometrical" structure by assigning vector spaces and linear maps to nodes and edges. Graph Neural Networks (GNNs) implicitly assume a graph with a trivial underlying sheaf. This choice is reflected in the structure of the graph Laplacian operator, the properties of the associated diffusion equation, and the characteristics of the convolutional models that discretise this equation. In this paper, we use cellular sheaf theory to show that the underlying geometry of the graph is deeply linked with the performance of GNNs in heterophilic settings and their oversmoothing behaviour. By considering a hierarchy of increasingly general sheaves, we study how the ability of the sheaf diffusion process to achieve linear separation of the classes in the infinite time limit expands. At the same time, we prove that when the sheaf is non-trivial, discretised parametric diffusion processes have greater control than GNNs over their asymptotic behaviour. On the practical side, we study how sheaves can be learned from data. The resulting sheaf diffusion models have many desirable properties that address the limitations of classical graph diffusion equations (and corresponding GNN models) and obtain state-of-the-art results in heterophilic settings. Overall, our work provides new connections between GNNs and algebraic topology and would be of interest to both fields.

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