  Directed networks such as gene regulation networks and neural networks are
connected by arcs (directed links). The nodes in a directed network are often
strongly interwound by a huge number of directed cycles, which lead to complex
information-processing dynamics in the network and make it highly challenging
to infer the intrinsic direction of information flow. In this theoretical
paper, based on the principle of minimum-feedback, we explore the node
hierarchy of directed networks and distinguish feedforward and feedback arcs.
Nearly optimal node hierarchy solutions, which minimize the number of feedback
arcs from lower-level nodes to higher-level nodes, are constructed by
belief-propagation and simulated-annealing methods. For real-world networks, we
quantify the extent of feedback scarcity by comparison with the ensemble of
direction-randomized networks and identify the most important feedback arcs.
Our methods are also useful for visualizing directed networks.
