  Node clustering is a powerful tool in the analysis of networks. We introduce
a graph neural network framework, named DIGRAC, to obtain node embeddings for
directed networks in a self-supervised manner, including a novel probabilistic
imbalance loss, which can be used for network clustering. Here, we propose
\textit{directed flow imbalance} measures, which are tightly related to
directionality, to reveal clusters in the network even when there is no density
difference between clusters. In contrast to standard approaches in the
literature, in this paper, directionality is not treated as a nuisance, but
rather contains the main signal. DIGRAC optimizes directed flow imbalance for
clustering without requiring label supervision, unlike existing graph neural
network methods, and can naturally incorporate node features, unlike existing
spectral methods. Extensive experimental results on synthetic data, in the form
of directed stochastic block models, and real-world data at different scales,
demonstrate that our method, based on flow imbalance, attains state-of-the-art
results on directed graph clustering when compared against 10 state-of-the-art
methods from the literature, for a wide range of noise and sparsity levels,
graph structures, and topologies, and even outperforms supervised methods.
