Graph neural networks (GNNs) are widely used in domains like social networks and biological systems. However, the locality assumption of GNNs, which limits information exchange to neighboring nodes, hampers their ability to capture long-range dependencies and global patterns in graphs. To address this, we propose a new inductive bias based on variational analysis, drawing inspiration from the Brachistochrone problem. Our framework establishes a mapping between discrete GNN models and continuous diffusion functionals. This enables the design of application-specific objective functions in the continuous domain and the construction of discrete deep models with mathematical guarantees. To tackle over-smoothing in GNNs, we analyze the existing layer-by-layer graph embedding models and identify that they are equivalent to l2-norm integral functionals of graph gradients, which cause over-smoothing. Similar to edge-preserving filters in image denoising, we introduce total variation (TV) to align the graph diffusion pattern with global community topologies. Additionally, we devise a selective mechanism to address the trade-off between model depth and over-smoothing, which can be easily integrated into existing GNNs. Furthermore, we propose a novel generative adversarial network (GAN) that predicts spreading flows in graphs through a neural transport equation. To mitigate vanishing flows, we customize the objective function to minimize transportation within each community while maximizing inter-community flows. Our GNN models achieve state-of-the-art (SOTA) performance on popular graph learning benchmarks such as Cora, Citeseer, and Pubmed.
High-order Graph Neural Networks (HO-GNNs) have been developed to infer consistent latent spaces in the heterophilic regime, where the label distribution is not correlated with the graph structure. However, most of the existing HO-GNNs are hop-based, i.e., they rely on the powers of the transition matrix. As a result, these architectures are not fully reactive to the classification loss and the achieved structural filters have static supports. In other words, neither the filters' supports nor their coefficients can be learned with these networks. They are confined, instead, to learn combinations of filters. To address the above concerns, we propose Diffusion-jump GNNs a method relying on asymptotic diffusion distances that operates on jumps. A diffusion-pump generates pairwise distances whose projections determine both the support and coefficients of each structural filter. These filters are called jumps because they explore a wide range of scales in order to find bonds between scattered nodes with the same label. Actually, the full process is controlled by the classification loss. Both the jumps and the diffusion distances react to classification errors (i.e. they are learnable). Homophiliation, i.e., the process of learning piecewise smooth latent spaces in the heterophilic regime, is formulated as a Dirichlet problem: the known labels determine the border nodes and the diffusion-pump ensures a minimal deviation of the semi-supervised grouping from a canonical unsupervised grouping. This triggers the update of both the diffusion distances and, consequently, the jumps in order to minimize the classification error. The Dirichlet formulation has several advantages. It leads to the definition of structural heterophily, a novel measure beyond edge heterophily. It also allows us to investigate links with (learnable) diffusion distances, absorbing random walks and stochastic diffusion.
We propose Unsupervised Episode Generation method called Neighbors as Queries (NaQ) to solve the Few-Shot Node-Classification (FSNC) task by unsupervised Graph Meta-learning. Doing so enables full utilization of the information of all nodes in a graph, which is not possible in current supervised meta-learning methods for FSNC due to the label-scarcity problem. In addition, unlike unsupervised Graph Contrastive Learning (GCL) methods that overlook the downstream task to be solved at the training phase resulting in vulnerability to class imbalance of a graph, we adopt the episodic learning framework that allows the model to be aware of the downstream task format, i.e., FSNC. The proposed NaQ is a simple but effective unsupervised episode generation method that randomly samples nodes from a graph to make a support set, followed by similarity-based sampling of nodes to make the corresponding query set. Since NaQ is model-agnostic, any existing supervised graph meta-learning methods can be trained in an unsupervised manner, while not sacrificing much of their performance or sometimes even improving them. Extensive experimental results demonstrate the effectiveness of our proposed unsupervised episode generation method for graph meta-learning towards the FSNC task. Our code is available at: https://github.com/JhngJng/NaQ-PyTorch.
Few-shot node classification, which aims to predict labels for nodes on graphs with only limited labeled nodes as references, is of great significance in real-world graph mining tasks. Particularly, in this paper, we refer to the task of classifying nodes in classes with a few labeled nodes as the few-shot node classification problem. To tackle such a label shortage issue, existing works generally leverage the meta-learning framework, which utilizes a number of episodes to extract transferable knowledge from classes with abundant labeled nodes and generalizes the knowledge to other classes with limited labeled nodes. In essence, the primary aim of few-shot node classification is to learn node embeddings that are generalizable across different classes. To accomplish this, the GNN encoder must be able to distinguish node embeddings between different classes, while also aligning embeddings for nodes in the same class. Thus, in this work, we propose to consider both the intra-class and inter-class generalizability of the model. We create a novel contrastive meta-learning framework on graphs, named COSMIC, with two key designs. First, we propose to enhance the intra-class generalizability by involving a contrastive two-step optimization in each episode to explicitly align node embeddings in the same classes. Second, we strengthen the inter-class generalizability by generating hard node classes via a novel similarity-sensitive mix-up strategy. Extensive experiments on few-shot node classification datasets verify the superiority of our framework over state-of-the-art baselines. Our code is provided at https://github.com/SongW-SW/COSMIC.
Graph convolutional networks (GCNs) have achieved huge success in several machine learning (ML) tasks on graph-structured data. Recently, several sampling techniques have been proposed for the efficient training of GCNs and to improve the performance of GCNs on ML tasks. Specifically, the subgraph-based sampling approaches such as ClusterGCN and GraphSAINT have achieved state-of-the-art performance on the node classification tasks. These subgraph-based sampling approaches rely on heuristics -- such as graph partitioning via edge cuts -- to identify clusters that are then treated as minibatches during GCN training. In this work, we hypothesize that rather than relying on such heuristics, one can learn a reinforcement learning (RL) policy to compute efficient clusters that lead to effective GCN performance. To that end, we propose PolicyClusterGCN, an online RL framework that can identify good clusters for GCN training. We develop a novel Markov Decision Process (MDP) formulation that allows the policy network to predict ``importance" weights on the edges which are then utilized by a clustering algorithm (Graclus) to compute the clusters. We train the policy network using a standard policy gradient algorithm where the rewards are computed from the classification accuracies while training GCN using clusters given by the policy. Experiments on six real-world datasets and several synthetic datasets show that PolicyClusterGCN outperforms existing state-of-the-art models on node classification task.
Predicting node labels on a given graph is a widely studied problem with many applications, including community detection and molecular graph prediction. This paper considers predicting multiple node labeling functions on graphs simultaneously and revisits this problem from a multitask learning perspective. For a concrete example, consider overlapping community detection: each community membership is a binary node classification task. Due to complex overlapping patterns, we find that negative transfer is prevalent when we apply naive multitask learning to multiple community detection, as task relationships are highly nonlinear across different node labeling. To address the challenge, we develop an algorithm to cluster tasks into groups based on a higher-order task affinity measure. We then fit a multitask model on each task group, resulting in a boosting procedure on top of the baseline model. We estimate the higher-order task affinity measure between two tasks as the prediction loss of one task in the presence of another task and a random subset of other tasks. Then, we use spectral clustering on the affinity score matrix to identify task grouping. We design several speedup techniques to compute the higher-order affinity scores efficiently and show that they can predict negative transfers more accurately than pairwise task affinities. We validate our procedure using various community detection and molecular graph prediction data sets, showing favorable results compared with existing methods. Lastly, we provide a theoretical analysis to show that under a planted block model of tasks on graphs, our affinity scores can provably separate tasks into groups.
While graph heterophily has been extensively studied in recent years, a fundamental research question largely remains nascent: How and to what extent will graph heterophily affect the prediction performance of graph neural networks (GNNs)? In this paper, we aim to demystify the impact of graph heterophily on GNN spectral filters. Our theoretical results show that it is essential to design adaptive polynomial filters that adapts different degrees of graph heterophily to guarantee the generalization performance of GNNs. Inspired by our theoretical findings, we propose a simple yet powerful GNN named GPatcher by leveraging the MLP-Mixer architectures. Our approach comprises two main components: (1) an adaptive patch extractor function that automatically transforms each node's non-Euclidean graph representations to Euclidean patch representations given different degrees of heterophily, and (2) an efficient patch mixer function that learns salient node representation from both the local context information and the global positional information. Through extensive experiments, the GPatcher model demonstrates outstanding performance on node classification compared with popular homophily GNNs and state-of-the-art heterophily GNNs.
Geometric deep learning (GDL) models have demonstrated a great potential for the analysis of non-Euclidian data. They are developed to incorporate the geometric and topological information of non-Euclidian data into the end-to-end deep learning architectures. Motivated by the recent success of discrete Ricci curvature in graph neural network (GNNs), we propose TorGNN, an analytic Torsion enhanced Graph Neural Network model. The essential idea is to characterize graph local structures with an analytic torsion based weight formula. Mathematically, analytic torsion is a topological invariant that can distinguish spaces which are homotopy equivalent but not homeomorphic. In our TorGNN, for each edge, a corresponding local simplicial complex is identified, then the analytic torsion (for this local simplicial complex) is calculated, and further used as a weight (for this edge) in message-passing process. Our TorGNN model is validated on link prediction tasks from sixteen different types of networks and node classification tasks from three types of networks. It has been found that our TorGNN can achieve superior performance on both tasks, and outperform various state-of-the-art models. This demonstrates that analytic torsion is a highly efficient topological invariant in the characterization of graph structures and can significantly boost the performance of GNNs.
It has been discovered that Graph Convolutional Networks (GCNs) encounter a remarkable drop in performance when multiple layers are piled up. The main factor that accounts for why deep GCNs fail lies in over-smoothing, which isolates the network output from the input with the increase of network depth, weakening expressivity and trainability. In this paper, we start by investigating refined measures upon DropEdge -- an existing simple yet effective technique to relieve over-smoothing. We term our method as DropEdge++ for its two structure-aware samplers in contrast to DropEdge: layer-dependent sampler and feature-dependent sampler. Regarding the layer-dependent sampler, we interestingly find that increasingly sampling edges from the bottom layer yields superior performance than the decreasing counterpart as well as DropEdge. We theoretically reveal this phenomenon with Mean-Edge-Number (MEN), a metric closely related to over-smoothing. For the feature-dependent sampler, we associate the edge sampling probability with the feature similarity of node pairs, and prove that it further correlates the convergence subspace of the output layer with the input features. Extensive experiments on several node classification benchmarks, including both full- and semi- supervised tasks, illustrate the efficacy of DropEdge++ and its compatibility with a variety of backbones by achieving generally better performance over DropEdge and the no-drop version.
Graph Pooling technology plays an important role in graph node classification tasks. Sorting pooling technologies maintain large-value units for pooling graphs of varying sizes. However, by analyzing the statistical characteristic of activated units after pooling, we found that a large number of units dropped by sorting pooling are negative-value units that contain useful information and can contribute considerably to the final decision. To maintain more useful information, a novel pooling technology, called Geometric Pooling (GP), was proposed to contain the unique node features with negative values by measuring the similarity of all node features. We reveal the effectiveness of GP from the entropy reduction view. The experiments were conducted on TUdatasets to show the effectiveness of GP. The results showed that the proposed GP outperforms the SOTA graph pooling technologies by 1%\sim5% with fewer parameters.
Fair graph partition of social networks is a crucial step toward ensuring fair and non-discriminatory treatments in unsupervised user analysis. Current fair partition methods typically consider node balance, a notion pursuing a proportionally balanced number of nodes from all demographic groups, but ignore the bias induced by imbalanced edges in each cluster. To address this gap, we propose a notion edge balance to measure the proportion of edges connecting different demographic groups in clusters. We analyze the relations between node balance and edge balance, then with line graph transformations, we propose a co-embedding framework to learn dual node and edge fairness-aware representations for graph partition. We validate our framework through several social network datasets and observe balanced partition in terms of both nodes and edges along with good utility. Moreover, we demonstrate our fair partition can be used as pseudo labels to facilitate graph neural networks to behave fairly in node classification and link prediction tasks.
Contrastive learning methods have attracted considerable attention due to their remarkable success in analyzing graph-structured data. Inspired by the success of contrastive learning, we propose a novel framework for contrastive disentangled learning on graphs, employing a disentangled graph encoder and two carefully crafted self-supervision signals. Specifically, we introduce a disentangled graph encoder to enforce the framework to distinguish various latent factors corresponding to underlying semantic information and learn the disentangled node embeddings. Moreover, to overcome the heavy reliance on labels, we design two self-supervision signals, namely node specificity and channel independence, which capture informative knowledge without the need for labeled data, thereby guiding the automatic disentanglement of nodes. Finally, we perform node classification tasks on three citation networks by using the disentangled node embeddings, and the relevant analysis is provided. Experimental results validate the effectiveness of the proposed framework compared with various baselines.
Hypergraphs are a powerful abstraction for representing higher-order interactions between entities of interest. To exploit these relationships in making downstream predictions, a variety of hypergraph neural network architectures have recently been proposed, in large part building upon precursors from the more traditional graph neural network (GNN) literature. Somewhat differently, in this paper we begin by presenting an expressive family of parameterized, hypergraph-regularized energy functions. We then demonstrate how minimizers of these energies effectively serve as node embeddings that, when paired with a parameterized classifier, can be trained end-to-end via a supervised bilevel optimization process. Later, we draw parallels between the implicit architecture of the predictive models emerging from the proposed bilevel hypergraph optimization, and existing GNN architectures in common use. Empirically, we demonstrate state-of-the-art results on various hypergraph node classification benchmarks. Code is available at https://github.com/yxzwang/PhenomNN.
Class imbalance is the phenomenon that some classes have much fewer instances than others, which is ubiquitous in real-world graph-structured scenarios. Recent studies find that off-the-shelf Graph Neural Networks (GNNs) would under-represent minor class samples. We investigate this phenomenon and discover that the subspaces of minor classes being squeezed by those of the major ones in the latent space is the main cause of this failure. We are naturally inspired to enlarge the decision boundaries of minor classes and propose a general framework GraphSHA by Synthesizing HArder minor samples. Furthermore, to avoid the enlarged minor boundary violating the subspaces of neighbor classes, we also propose a module called SemiMixup to transmit enlarged boundary information to the interior of the minor classes while blocking information propagation from minor classes to neighbor classes. Empirically, GraphSHA shows its effectiveness in enlarging the decision boundaries of minor classes, as it outperforms various baseline methods in class-imbalanced node classification with different GNN backbone encoders over seven public benchmark datasets. Code is avilable at https://github.com/wenzhilics/GraphSHA.
The rise of cryptocurrencies like Bitcoin, which enable transactions with a degree of pseudonymity, has led to a surge in various illicit activities, including ransomware payments and transactions on darknet markets. These illegal activities often utilize Bitcoin as the preferred payment method. However, current tools for detecting illicit behavior either rely on a few heuristics and laborious data collection processes or employ computationally inefficient graph neural network (GNN) models that are challenging to interpret. To overcome the computational and interpretability limitations of existing techniques, we introduce an effective solution called Chainlet Orbits. This approach embeds Bitcoin addresses by leveraging their topological characteristics in transactions. By employing our innovative address embedding, we investigate e-crime in Bitcoin networks by focusing on distinctive substructures that arise from illicit behavior. The results of our node classification experiments demonstrate superior performance compared to state-of-the-art methods, including both topological and GNN-based approaches. Moreover, our approach enables the use of interpretable and explainable machine learning models in as little as 15 minutes for most days on the Bitcoin transaction network.
Node classification is one of the hottest tasks in graph analysis. Though existing studies have explored various node representations in directed and undirected graphs, they have overlooked the distinctions of their capabilities to capture the information of graphs. To tackle the limitation, we investigate various combinations of node representations (aggregated features vs. adjacency lists) and edge direction awareness within an input graph (directed vs. undirected). We address the first empirical study to benchmark the performance of various GNNs that use either combination of node representations and edge direction awareness. Our experiments demonstrate that no single combination stably achieves state-of-the-art results across datasets, which indicates that we need to select appropriate combinations depending on the dataset characteristics. In response, we propose a simple yet holistic classification method A2DUG which leverages all combinations of node representations in directed and undirected graphs. We demonstrate that A2DUG stably performs well on various datasets and improves the accuracy up to 11.29 compared with the state-of-the-art methods. To spur the development of new methods, we publicly release our complete codebase under the MIT license.
Graph neural networks have shown impressive capabilities in solving various graph learning tasks, particularly excelling in node classification. However, their effectiveness can be hindered by the challenges arising from the widespread existence of noisy measurements associated with the topological or nodal information present in real-world graphs. These inaccuracies in observations can corrupt the crucial patterns within the graph data, ultimately resulting in undesirable performance in practical applications. To address these issues, this paper proposes a novel uncertainty-aware graph learning framework motivated by distributionally robust optimization. Specifically, we use a graph neural network-based encoder to embed the node features and find the optimal node embeddings by minimizing the worst-case risk through a minimax formulation. Such an uncertainty-aware learning process leads to improved node representations and a more robust graph predictive model that effectively mitigates the impact of uncertainty arising from data noise. Our experimental result shows that the proposed framework achieves superior predictive performance compared to the state-of-the-art baselines under various noisy settings.
Meta-learning has emerged as a powerful training strategy for few-shot node classification, demonstrating its effectiveness in the transductive setting. However, the existing literature predominantly focuses on transductive few-shot node classification, neglecting the widely studied inductive setting in the broader few-shot learning community. This oversight limits our comprehensive understanding of the performance of meta-learning based methods on graph data. In this work, we conduct an empirical study to highlight the limitations of current frameworks in the inductive few-shot node classification setting. Additionally, we propose a simple yet competitive baseline approach specifically tailored for inductive few-shot node classification tasks. We hope our work can provide a new path forward to better understand how the meta-learning paradigm works in the graph domain.
When training a Neural Network, it is optimized using the available training data with the hope that it generalizes well to new or unseen testing data. At the same absolute value, a flat minimum in the loss landscape is presumed to generalize better than a sharp minimum. Methods for determining flat minima have been mostly researched for independent and identically distributed (i. i. d.) data such as images. Graphs are inherently non-i. i. d. since the vertices are edge-connected. We investigate flat minima methods and combinations of those methods for training graph neural networks (GNNs). We use GCN and GAT as well as extend Graph-MLP to work with more layers and larger graphs. We conduct experiments on small and large citation, co-purchase, and protein datasets with different train-test splits in both the transductive and inductive training procedure. Results show that flat minima methods can improve the performance of GNN models by over 2 points, if the train-test split is randomized. Following Shchur et al., randomized splits are essential for a fair evaluation of GNNs, as other (fixed) splits like 'Planetoid' are biased. Overall, we provide important insights for improving and fairly evaluating flat minima methods on GNNs. We recommend practitioners to always use weight averaging techniques, in particular EWA when using early stopping. While weight averaging techniques are only sometimes the best performing method, they are less sensitive to hyperparameters, need no additional training, and keep the original model unchanged. All source code is available in https://github.com/Foisunt/FMMs-in-GNNs.
Network embedding, a graph representation learning method illustrating network topology by mapping nodes into lower-dimension vectors, is challenging to accommodate the ever-changing dynamic graphs in practice. Existing research is mainly based on node-by-node embedding modifications, which falls into the dilemma of efficient calculation and accuracy. Observing that the embedding dimensions are usually much smaller than the number of nodes, we break this dilemma with a novel dynamic network embedding paradigm that rotates and scales the axes of embedding space instead of a node-by-node update. Specifically, we propose the Dynamic Adjacency Matrix Factorization (DAMF) algorithm, which achieves an efficient and accurate dynamic network embedding by rotating and scaling the coordinate system where the network embedding resides with no more than the number of edge modifications changes of node embeddings. Moreover, a dynamic Personalized PageRank is applied to the obtained network embeddings to enhance node embeddings and capture higher-order neighbor information dynamically. Experiments of node classification, link prediction, and graph reconstruction on different-sized dynamic graphs suggest that DAMF advances dynamic network embedding. Further, we unprecedentedly expand dynamic network embedding experiments to billion-edge graphs, where DAMF updates billion-level parameters in less than 10ms.