Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework. Specifically, we propose fractional graph Laplacian neural ODEs, which describe non-local dynamics. We prove that our approach allows propagating information between distant nodes while maintaining a low probability of long-distance jumps. Moreover, we show that our method is more flexible with respect to the convergence of the graph's Dirichlet energy, thereby mitigating oversmoothing. We conduct extensive experiments on synthetic and real-world graphs, both directed and undirected, demonstrating our method's versatility across diverse graph homophily levels.

In this paper, we study the factors that contribute to the effect of oversmoothing in deep Graph Neural Networks (GNNs). Specifically, our analysis is based on a new metric (Mean Average Squared Distance - $MASED$) to quantify the extent of oversmoothing. We derive layer-wise bounds on $MASED$, which aggregate to yield global upper and lower distance bounds. Based on this quantification of oversmoothing, we further analyze the importance of two different properties of the model; namely the norms of the generated node embeddings, along with the largest and smallest singular values of the weight matrices. Building on the insights drawn from the theoretical analysis, we show that oversmoothing increases as the number of trainable weight matrices and the number of adjacency matrices increases. We also use the derived layer-wise bounds on $MASED$ to form a proposal for decoupling the number of hops (i.e., adjacency depth) from the number of weight matrices. In particular, we introduce G-Reg, a regularization scheme that increases the bounds, and demonstrate through extensive experiments that by doing so node classification accuracy increases, achieving robustness at large depths. We further show that by reducing oversmoothing in deep networks, we can achieve better results in some tasks than using shallow ones. Specifically, we experiment with a ``cold start" scenario, i.e., when there is no feature information for the unlabeled nodes. Finally, we show empirically the trade-off between receptive field size (i.e., number of weight matrices) and performance, using the $MASED$ bounds. This is achieved by distributing adjacency hops across a small number of trainable layers, avoiding the extremes of under- or over-parameterization of the GNN.

Oversmoothing has long been identified as a major limitation of Graph Neural Networks (GNNs): input node features are smoothed at each layer and converge to a non-informative representation, if the weights of the GNN are sufficiently bounded. This assumption is crucial: if, on the contrary, the weights are sufficiently large, then oversmoothing may not happen. Theoretically, GNN could thus learn to not oversmooth. However it does not really happen in practice, which prompts us to examine oversmoothing from an optimization point of view. In this paper, we analyze backward oversmoothing, that is, the notion that backpropagated errors used to compute gradients are also subject to oversmoothing from output to input. With non-linear activation functions, we outline the key role of the interaction between forward and backward smoothing. Moreover, we show that, due to backward oversmoothing, GNNs provably exhibit many spurious stationary points: as soon as the last layer is trained, the whole GNN is at a stationary point. As a result, we can exhibit regions where gradients are near-zero while the loss stays high. The proof relies on the fact that, unlike forward oversmoothing, backward errors are subjected to a linear oversmoothing even in the presence of non-linear activation function, such that the average of the output error plays a key role. Additionally, we show that this phenomenon is specific to deep GNNs, and exhibit counter-example Multi-Layer Perceptron. This paper is a step toward a more complete comprehension of the optimization landscape specific to GNNs.

In contrast to classes of neural networks where the learned representations become increasingly expressive with network depth, the learned representations in graph neural networks (GNNs), tend to become increasingly similar. This phenomena, known as oversmoothing, is characterized by learned representations that cannot be reliably differentiated leading to reduced predictive performance. In this paper, we propose an analogy between oversmoothing in GNNs and consensus or agreement in opinion dynamics. Through this analogy, we show that the message passing structure of recent continuous-depth GNNs is equivalent to a special case of opinion dynamics (i.e., linear consensus models) which has been theoretically proven to converge to consensus (i.e., oversmoothing) for all inputs. Using the understanding developed through this analogy, we design a new continuous-depth GNN model based on nonlinear opinion dynamics and prove that our model, which we call behavior-inspired message passing neural network (BIMP) circumvents oversmoothing for general inputs. Through extensive experiments, we show that BIMP is robust to oversmoothing and adversarial attack, and consistently outperforms competitive baselines on numerous benchmarks.

Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework.

This paper presents a novel study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Diverging from extant approaches grounded in random walk analysis or particle systems, we approach this problem through operator semigroup theory. This theoretical framework allows us to rigorously prove that oversmoothing is intrinsically linked to the ergodicity of the diffusion operator. This finding further poses a general and mild ergodicity-breaking condition, encompassing the various specific solutions previously offered, thereby presenting a more universal and theoretically grounded approach to mitigating oversmoothing in diffusion-based GNNs. Additionally, we offer a probabilistic interpretation of our theory, forging a link with prior works and broadening the theoretical horizon. Our experimental results reveal that this ergodicity-breaking term effectively mitigates oversmoothing measured by Dirichlet energy, and simultaneously enhances performance in node classification tasks.

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.