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I'm new to the graph convolution network. I wonder what is the main purpose of applying data with graph structure to CNN?

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There are some problems that involve graphs and manifolds (sometimes collectively called non-Euclidean data), such as molecule design and generation, drug repositioning, social networks analysis, brain imaging, fake news detection, recommender systems, neutrino detection, computer vision and graphics and shape (e.g. hand or face) completion or generation (generative models).

The main benefit of geometric deep learning (deep learning applied to graphs and manifolds) is that you do not lose the information encoded in the graphs (or manifolds), which, otherwise, you would likely lose because you would need to convert your graphs (or manifolds) to an equivalent vector representation that you can feed into the existing CNN or other standard neural networks.

Note that you cannot directly apply the usual convolution operation to graphs, because, for example, graphs do not have the notion of relative positions of the nodes. Furthermore, note that graph networks have little to do with CNNs, even if they are sometimes called graph convolution networks.

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