### In-Person Poster presentation / top 25% paper

## Effects of Graph Convolutions in Multi-layer Networks

### Aseem Baranwal · Kimon Fountoulakis · Aukosh Jagannath

##### MH1-2-3-4 #140

Keywords: [ Theory ] [ contextual stochastic block model ] [ classification threshold ] [ node classification ] [ graph neural networks ]

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 $1/\sqrt[4]{\rm deg}$, where ${\rm deg}$ 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 neural 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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