标题： 有向网络中的反馈弧与节点层次结构 显示英文标题

标题： Feedback arcs and node hierarchy in directed networks

摘要： 诸如基因调控网络和神经网络之类的有向网络由弧（有向链接）连接。 有向网络中的节点通常通过大量的有向环紧密交织在一起，这导致网络中复杂的信息处理动态，并使得推断信息流的固有方向变得极具挑战性。 在这篇理论论文中，基于最小反馈原则，我们探索了有向网络的节点层次结构，并区分了前馈弧和反馈弧。 通过置信传播和模拟退火方法，构建了几乎最优的节点层次结构解决方案，这些方案最小化了从低层节点到高层节点的反馈弧数量。 对于真实世界中的网络，我们通过与方向随机化的网络集合进行比较来量化反馈稀缺的程度，并识别最重要的反馈弧。 我们的方法也有助于有向网络的可视化。 显示英文摘要

摘要： 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.