Disentangling large-scale supply networks: f-HiCoNE framework for flow-hierarchical clustering via combinatorial Hodge decomposition

Modern society relies on complex supply chains to sustain the flow of goods and services that are essential to daily life. While traditional supply chain theory assumes a clear, hierarchical flow from upstream suppliers to downstream customers, observable real-world transaction networks rarely exhibit this acyclic structure. Instead, detailed inter-firm data reveal that interwoven networks are heavily entangled by cyclic flows. Consequently, without appropriate partitioning of these massive inter-firm networks, the latent flow-hierarchical structures that are central to supply chain concepts remain obscure. To address this analytical challenge, we introduce the flow-Hierarchical Community Network Extraction (f-HiCoNE) framework. By applying combinatorial Hodge decomposition, this approach disentangles the complex inter-firm network by isolating the acyclic gradient flow to quantify the flow-hierarchical parts and partition the graph. By applying f-HiCoNE to a nationwide transaction dataset of approximately 650,000 firms, we extracted clusters exhibiting the flow-hierarchical organisation that supply-chain theory assumes. These clusters demonstrated strong flow-hierarchical organisation, wherein the upstream-downstream positioning of firms was well described by local scalar potentials, revealing distinct geographically localised industrial ecosystems. This study provides a map that could help firms understand their surrounding environment and locate their position within an inter-firm network and opens a new research avenue focused on flow-hierarchy clustering in supply chain analysis.