  Graph embedding provides a feasible methodology to conduct pattern
classification for graph-structured data by mapping each data into the
vectorial space. Various pioneering works are essentially coding method that
concentrates on a vectorial representation about the inner properties of a
graph in terms of the topological constitution, node attributions, link
relations, etc. However, the classification for each targeted data is a
qualitative issue based on understanding the overall discrepancies within the
dataset scale. From the statistical point of view, these discrepancies manifest
a metric distribution over the dataset scale if the distance metric is adopted
to measure the pairwise similarity or dissimilarity. Therefore, we present a
novel embedding strategy named $\mathbf{MetricDistribution2vec}$ to extract
such distribution characteristics into the vectorial representation for each
data. We demonstrate the application and effectiveness of our representation
method in the supervised prediction tasks on extensive real-world structural
graph datasets. The results have gained some unexpected increases compared with
a surge of baselines on all the datasets, even if we take the lightweight
models as classifiers. Moreover, the proposed methods also conducted
experiments in Few-Shot classification scenarios, and the results still show
attractive discrimination in rare training samples based inference.
