Directionality and node heterogeneity reshape criticality in hypergraph percolation

Directed and heterogeneous hypergraphs capture directional higher-order interactions with intrinsically asymmetric functional dependencies among nodes. As a result, damage to certain nodes can suppress entire hyperedges, whereas failure of others only weakens interactions. Metabolic reaction networks offer an intuitive example of such asymmetric dependencies. Here we develop a message-passing and statistical mechanics framework for percolation in directed hypergraphs that explicitly incorporates directionality and node heterogeneity. Remarkably, we show that these hypergraph features have a fundamental effect on the critical properties of hypergraph percolation, reshaping criticality in a way that depends on network structure. Specifically, we derive anomalous critical exponents that depend on whether node or hyperedge percolation is considered in maximally correlated, heavy-tailed regimes. These theoretical predictions are validated on synthetic hypergraph models and on a real directed metabolic network, opening new perspectives for the characterization of the robustness and resilience of real-world directed, heterogeneous higher-order networks.

💡 Research Summary

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The paper introduces a novel percolation framework for directed hypergraphs that explicitly incorporates two key real‑world features: (i) the intrinsic directionality of higher‑order interactions (input‑to‑output flow) and (ii) node heterogeneity in the form of “anchor” nodes whose failure disables an entire hyperedge. By mapping a directed hypergraph onto a bipartite factor graph, the authors derive forward and backward message‑passing equations that capture the emergence of three giant components: the Hypergraph Giant Out Component (HGOUT), the Hypergraph Giant In Component (HGIN), and the Hypergraph Giant Strongly Connected Component (HGSCC). Forward messages propagate along the prescribed direction and determine reachability for HGOUT; backward messages propagate against the direction and determine HGIN. A node (or hyperedge) belongs to the HGSCC only when it is reachable both forward and backward, i.e., when the two message streams intersect.

The probability that a node is an anchor is denoted by θ. Surviving nodes have probability p_N, while hyperedges survive with probability p_H