Effects of Graph Convolutions in Deep Networks

• URL: http://arxiv.org/abs/2204.09297v1
• Date: Wed, 20 Apr 2022 08:24:43 GMT
• Title: Effects of Graph Convolutions in Deep Networks
• Authors: Aseem Baranwal, Kimon Fountoulakis, Aukosh Jagannath
• Abstract summary: We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data. We provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network.
• Score: 8.937905773981702
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.

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