标题： 网络的循环方向流分解 显示英文标题

标题： Circular Directional Flow Decomposition of Networks

摘要： 我们介绍了圆方向流分解（CDFD），这是一种用于分析加权有向网络循环性的新框架。 CDFD 将流量分为两个组成部分：一个循环（无散度）分量和一个携带所有净方向流量的非循环分量。 这产生了一个归一化的循环性指数，范围从 0（完全非循环）到 1（由纯循环叠加形成的网络），其补值衡量方向性。 该指数捕捉了参与循环的流量比例，并具有多种解释——例如系统闭合、反馈、加权强连通性、结构冗余或低效。 尽管分解通常是非唯一的，但我们表明，所有分解的集合形成一个具有良好拓扑性质的几何空间。 在这个空间内，我们强调了两种与不同分析目标一致的基准分解：最小化净流量的最大循环性解和平衡流转发（BFF）解，后者是一种独特的、局部可计算的分解方式，按照原始网络结构的比例将循环流分布在所有可行的循环中。 我们在合成和经验网络上展示了这两种分解的解释价值和计算可行性。 它们在检测有意义的结构性变化方面优于现有的循环性指标。 分解还支持结构分析——例如映射循环流分布——并支持需要明确流分配或路由的实际应用，包括多方抵消和高效运输。 显示英文摘要

摘要： We introduce the Circular Directional Flow Decomposition (CDFD), a new framework for analyzing circularity in weighted directed networks. CDFD separates flow into two components: a circular (divergence-free) component and an acyclic component that carries all nett directional flow. This yields a normalized circularity index between 0 (fully acyclic) and 1 (for networks formed solely by the superposition of cycles), with the complement measuring directionality. This index captures the proportion of flow involved in cycles, and admits a range of interpretations - such as system closure, feedback, weighted strong connectivity, structural redundancy, or inefficiency. Although the decomposition is generally non-unique, we show that the set of all decompositions forms a well-structured geometric space with favourable topological properties. Within this space, we highlight two benchmark decompositions aligned with distinct analytical goals: the maximum circularity solution, which minimizes nett flow, and the Balanced Flow Forwarding (BFF) solution, a unique, locally computable decomposition that distributes circular flow across all feasible cycles in proportion to the original network structure. We demonstrate the interpretive value and computational tractability of both decompositions on synthetic and empirical networks. They outperform existing circularity metrics in detecting meaningful structural variation. The decomposition also enables structural analysis - such as mapping the distribution of cyclic flow - and supports practical applications that require explicit flow allocation or routing, including multilateral netting and efficient transport.