  Learning distributed representations for nodes in graphs is a crucial
primitive in network analysis with a wide spectrum of applications. Linear
graph embedding methods learn such representations by optimizing the likelihood
of both positive and negative edges while constraining the dimension of the
embedding vectors. We argue that the generalization performance of these
methods is not due to the dimensionality constraint as commonly believed, but
rather the small norm of embedding vectors. Both theoretical and empirical
evidence are provided to support this argument: (a) we prove that the
generalization error of these methods can be bounded by limiting the norm of
vectors, regardless of the embedding dimension; (b) we show that the
generalization performance of linear graph embedding methods is correlated with
the norm of embedding vectors, which is small due to the early stopping of SGD
and the vanishing gradients. We performed extensive experiments to validate our
analysis and showcased the importance of proper norm regularization in
practice.
