  Graph neural networks (GNNs) extend convolutional neural networks (CNNs) to
graph-based data. A question that arises is how much performance improvement
does the underlying graph structure in the GNN provide over the CNN (that
ignores this graph structure). To address this question, we introduce edge
entropy and evaluate how good an indicator it is for possible performance
improvement of GNNs over CNNs. Our results on node classification with
synthetic and real datasets show that lower values of edge entropy predict
larger expected performance gains of GNNs over CNNs, and, conversely, higher
edge entropy leads to expected smaller improvement gains.
