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Register a new userImproving the Expressive Power of Graph Neural Network with Tinhofer Algorithm
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Added by
Alan JiaXiang Guo
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In recent years, Graph Neural Network (GNN) has bloomly progressed for its power in processing graph-based data. Most GNNs follow a message passing scheme, and their expressive power is mathematically limited by the discriminative ability of the Weisfeiler-Lehman (WL) test. Following Tinhofers research on compact graphs, we propose a variation of the message passing scheme, called the Weisfeiler-Lehman-Tinhofer GNN (WLT-GNN), that theoretically breaks through the limitation of the WL test. In addition, we conduct comparative experiments and ablation studies on several well-known datasets. The results show that the proposed methods have comparable performances and better expressive power on these datasets.
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Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to measure the expressiveness of graph neural networks (GNNs), showing that the neighborhood aggregation GNNs were at most as powerful as 1-WL test in distinguishing graph structures. There were also improvements proposed in analogy to $k$-WL test ($k>1$). However, the aggregators in these GNNs are far from injective as required by the WL test, and suffer from weak distinguishing strength, making it become expressive bottlenecks. In this paper, we improve the expressiveness by exploring powerful aggregators. We reformulate aggregation with the corresponding aggregation coefficient matrix, and then systematically analyze the requirements of the aggregation coefficient matrix for building more powerful aggregators and even injective aggregators. It can also be viewed as the strategy for preserving the rank of hidden features, and implies that basic aggregators correspond to a special case of low-rank transformations. We also show the necessity of applying nonlinear units ahead of aggregation, which is different from most aggregation-based GNNs. Based on our theoretical analysis, we develop two GNN layers, ExpandingConv and CombConv. Experimental results show that our models significantly boost performance, especially for large and densely connected graphs.
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exact formula for high activation degrees, as well as upper and lower bounds on layer widths in order for deep polynomials networks to fill the ambient functional space. We also present computational evidence that it is profitable in terms of expressiveness for layer widths to increase monotonically and then decrease monotonically. Finally, we link our study to favorable optimization properties when training weights, and we draw intriguing connections with tensor and polynomial decompositions.
Despite the wide application of Graph Convolutional Network (GCN), one major limitation is that it does not benefit from the increasing depth and suffers from the oversmoothing problem. In this work, we first characterize this phenomenon from the information-theoretic perspective and show that under certain conditions, the mutual information between the output after $l$ layers and the input of GCN converges to 0 exponentially with respect to $l$. We also show that, on the other hand, graph decomposition can potentially weaken the condition of such convergence rate, which enabled our analysis for GraphCNN. While different graph structures can only benefit from the corresponding decomposition, in practice, we propose an automatic connectivity-aware graph decomposition algorithm, DeGNN, to improve the performance of general graph neural networks. Extensive experiments on widely adopted benchmark datasets demonstrate that DeGNN can not only significantly boost the performance of corresponding GNNs, but also achieves the state-of-the-art performances.
Neural network forms the foundation of deep learning and numerous AI applications. Classical neural networks are fully connected, expensive to train and prone to overfitting. Sparse networks tend to have convoluted structure search, suboptimal performance and limited usage. We proposed the novel uniform sparse network (USN) with even and sparse connectivity within each layer. USN has one striking property that its performance is independent of the substantial topology variation and enormous model space, thus offers a search-free solution to all above mentioned issues of neural networks. USN consistently and substantially outperforms the state-of-the-art sparse network models in prediction accuracy, speed and robustness. It even achieves higher prediction accuracy than the fully connected network with only 0.55% parameters and 1/4 computing time and resources. Importantly, USN is conceptually simple as a natural generalization of fully connected network with multiple improvements in accuracy, robustness and scalability. USN can replace the latter in a range of applications, data types and deep learning architectures. We have made USN open source at https://github.com/datapplab/sparsenet.
Graph Neural Networks (GNNs) are efficient approaches to process graph-structured data. Modelling long-distance node relations is essential for GNN training and applications. However, conventional GNNs suffer from bad performance in modelling long-distance node relations due to limited-layer information propagation. Existing studies focus on building deep GNN architectures, which face the over-smoothing issue and cannot model node relations in particularly long distance. To address this issue, we propose to model long-distance node relations by simply relying on shallow GNN architectures with two solutions: (1) Implicitly modelling by learning to predict node pair relations (2) Explicitly modelling by adding edges between nodes that potentially have the same label. To combine our two solutions, we propose a model-agnostic training framework named HighwayGraph, which overcomes the challenge of insufficient labeled nodes by sampling node pairs from the training set and adopting the self-training method. Extensive experimental results show that our HighwayGraph achieves consistent and significant improvements over four representative GNNs on three benchmark datasets.
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