  The network embedding problem aims to map nodes that are similar to each
other to vectors in a Euclidean space that are close to each other. Like
centrality analysis (ranking) and community detection, network embedding is in
general considered as an ill-posed problem, and its solution may depend on a
person's view on this problem. In this book chapter, we adopt the framework of
sampled graphs that treat a person's view as a sampling method for a network.
The modularity for a sampled graph, called the generalized modularity in the
book chapter, is a similarity matrix that has a specific probabilistic
interpretation. One of the main contributions of this book chapter is to
propose using the generalized modularity matrix for network embedding and show
that the network embedding problem can be treated as a trace maximization
problem like the community detection problem. Our generalized modularity
embedding approach is very general and flexible. In particular, we show that
the Laplacian eigenmaps is a special case of our generalized modularity
embedding approach. Also, we show that dimensionality reduction can be done by
using a particular sampled graph. Various experiments are conducted on real
datasets to illustrate the effectiveness of our approach.
