Phase transitions and optimal algorithms in the semi-supervised classfications in graphs: from belief propagation to convolution neural networks
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By analyzing Bayesian inference of generative model for random networks with both relations (edges) and node features (discrete labels), we perform an asymptotically exact analysis of the semi-supervised classfication problems on graph-structured data using the cavity method of statistical physics. We unveil detectability phase transitions which put fundamental limit on ability of classfications for all possible algorithms. Our theory naturally converts to a message passing algorithm which works all the way down to the phase transition in the underlying generative model, and can be translated to a graph convolution neural network algorithm which greatly outperforms existing algorithms including popular graph neural networks in synthetic networks. When applied to real-world datasets, our algorithm achieves comparable performance with the state-of-the art algorithms. Our approach provides benchmark datasets with continuously tunable parameters and optimal results, which can be used to evaluate performance of exiting graph neural networks, and to find and understand their strengths and limitations. In particular, we observe that popular GCNs have sparsity issue and ovefitting issue on large synthetic benchmarks, we also show how to overcome the issues by combining strengths of our approach.
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