  Graph prediction problems prevail in data analysis and machine learning. The
inverse prediction problem, namely to infer input data from given output
labels, is of emerging interest in various applications. In this work, we
develop \textit{invertible graph neural network} (iGNN), a deep generative
model to tackle the inverse prediction problem on graphs by casting it as a
conditional generative task. The proposed model consists of an invertible
sub-network that maps one-to-one from data to an intermediate encoded feature,
which allows forward prediction by a linear classification sub-network as well
as efficient generation from output labels via a parametric mixture model. The
invertibility of the encoding sub-network is ensured by a Wasserstein-2
regularization which allows free-form layers in the residual blocks. The model
is scalable to large graphs by a factorized parametric mixture model of the
encoded feature and is computationally scalable by using GNN layers. The
existence of invertible flow mapping is backed by theories of optimal transport
and diffusion process, and we prove the expressiveness of graph convolution
layers to approximate the theoretical flows of graph data. The proposed iGNN
model is experimentally examined on synthetic data, including the example on
large graphs, and the empirical advantage is also demonstrated on
real-application datasets of solar ramping event data and traffic flow anomaly
detection.
