  Graph distance metric learning serves as the foundation for many graph
learning problems, e.g., graph clustering, graph classification and graph
matching. Existing research works on graph distance metric (or graph kernels)
learning fail to maintain the basic properties of such metrics, e.g.,
non-negative, identity of indiscernibles, symmetry and triangle inequality,
respectively. In this paper, we will introduce a new graph neural network based
distance metric learning approaches, namely GB-DISTANCE (GRAPH-BERT based
Neural Distance). Solely based on the attention mechanism, GB-DISTANCE can
learn graph instance representations effectively based on a pre-trained
GRAPH-BERT model. Different from the existing supervised/unsupervised metrics,
GB-DISTANCE can be learned effectively in a semi-supervised manner. In
addition, GB-DISTANCE can also maintain the distance metric basic properties
mentioned above. Extensive experiments have been done on several benchmark
graph datasets, and the results demonstrate that GB-DISTANCE can out-perform
the existing baseline methods, especially the recent graph neural network model
based graph metrics, with a significant gap in computing the graph distance.
