  Graph encoder embedding, a recent technique for graph data, offers speed and
scalability in producing vertex-level representations from binary graphs. In
this paper, we extend the applicability of this method to a general graph
model, which includes weighted graphs, distance matrices, and kernel matrices.
We prove that the encoder embedding satisfies the law of large numbers and the
central limit theorem on a per-observation basis. Under certain condition, it
achieves asymptotic normality on a per-class basis, enabling optimal
classification through discriminant analysis. These theoretical findings are
validated through a series of experiments involving weighted graphs, as well as
text and image data transformed into general graph representations using
appropriate distance metrics.
