  Graph node embedding aims at learning a vector representation for all nodes
given a graph. It is a central problem in many machine learning tasks (e.g.,
node classification, recommendation, community detection). The key problem in
graph node embedding lies in how to define the dependence to neighbors.
Existing approaches specify (either explicitly or implicitly) certain
dependencies on neighbors, which may lead to loss of subtle but important
structural information within the graph and other dependencies among neighbors.
This intrigues us to ask the question: can we design a model to give the
maximal flexibility of dependencies to each node's neighborhood. In this paper,
we propose a novel graph node embedding (named PINE) via a novel notion of
partial permutation invariant set function, to capture any possible dependence.
Our method 1) can learn an arbitrary form of the representation function from
the neighborhood, withour losing any potential dependence structures, and 2) is
applicable to both homogeneous and heterogeneous graph embedding, the latter of
which is challenged by the diversity of node types. Furthermore, we provide
theoretical guarantee for the representation capability of our method for
general homogeneous and heterogeneous graphs. Empirical evaluation results on
benchmark data sets show that our proposed PINE method outperforms the
state-of-the-art approaches on producing node vectors for various learning
tasks of both homogeneous and heterogeneous graphs.
