  We present our ongoing work on understanding the limitations of graph
convolutional networks (GCNs) as well as our work on generalizations of graph
convolutions for representing more complex node attribute dependencies. Based
on an analysis of GCNs with the help of the corresponding computation graphs,
we propose a generalization of existing GCNs where the aggregation operations
are (a) determined by structural properties of the local neighborhood graphs
and (b) not restricted to weighted averages. We show that the proposed approach
is strictly more expressive while requiring only a modest increase in the
number of parameters and computations. We also show that the proposed
generalization is identical to standard convolutional layers when applied to
regular grid graphs.
