How circular is a directed network? A flow decomposition approach

Abstract We introduce Circular Directional Flow Decomposition (CDFD), a new framework for analysing 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: 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. 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 and supports practical applications that require explicit flow allocation such as multilateral netting.