A number of problems can be formulated as prediction on graph-structured data. In this work, we generalize the convolution operator from regular grids to arbitrary graphs while avoiding the spectral domain, which allows us to handle graphs of varying size and connectivity. To move beyond a simple diffusion, filter weights are conditioned on the specific edge labels in the neighborhood of a vertex. Together with the proper choice of graph coarsening, we explore constructing deep neural networks for graph classification. In particular, we demonstrate the generality of our formulation in point cloud classification, where we set the new state of the art, and on a graph classification dataset, where we outperform other deep learning approaches. The source code is available at this https URL
To cover these important cases, we formulate our filtering approach in the spatial domain, where the limited complexity of evaluation and the localization property is provided by construction. The main challenge here is dealing with weight sharing among local neighborhoods @cite_44 , as the number of vertices adjacent to a particular vertex varies and their ordering is often not well definable.
@cite_44
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In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n ,w e form a random graphby replacing eachvertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G .T h e resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also randomizing given graphs. The main specific result that we prove here (Theorem 1) is that if   3 is the smallest vertex degree in G, then almost all n-covers of G are connected. In subsequent papers we will address oth er graphproperties, suchas girth , expansion and chromatic number.