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.

Previously, the controllability problem of a linear time-invariant dynamical system was mapped to the maximum matching (MM) problem on the bipartite representation of the underlying directed graph, and the sizes of MMs on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit. Here we present an alternative theory to estimate MM sizes based on the core percolation theory and the perfect matching of cores. Our theory is much more simplified and easily interpreted, and can estimate MM sizes on random graphs with or without symmetry between out-and in-degree distributions. Our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems.