Directionality and node heterogeneity reshape criticality in hypergraph percolation

TL;DR

This work develops a unified framework for percolation on directed hypergraphs with anchor nodes, capturing how directionality and mandatory participants reshape robustness. Using a message-passing scheme and mean-field theory, it derives the simultaneous emergence of the Hypergraph Giant In Component, Hypergraph Giant Out Component, and Hypergraph Giant Strongly Connected Component at a single threshold expressed by $\hat{\Lambda}=1$, with anchor density shifting the critical point via $\pi_N$. It predicts an additive law for the HGSCC exponent $\beta=\beta^{(+)}+\beta^{(-)}$ in broad classes, while revealing anomalous scaling in maximally correlated heavy-tailed topologies, where exponents depend on tail exponents $\gamma_q,\gamma_m$ and the presence of anchors. The framework is validated on synthetic directed hypergraphs and a real directed E. coli metabolic network, showing how functional constraints increase fragility yet enable smooth functional recovery, and demonstrating that anchor-induced regularization can restore standard universality even with heavy-tailed structure.

Abstract

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.

Figures (12)

• Figure 1: Directionality and functional dependencies in hypergraph percolation.(A) Representation of a directed hyperedge as a bipartite factor graph, where directionality encodes collective interactions from input nodes to output nodes. Anchor nodes (red) act as indispensable elements that control hyperedge activation, whereas non-anchor nodes (yellow) participate without determining functionality. (B) Anchor-node--induced functional dependency. Removal of an anchor node deactivates the entire hyperedge, while removal of a non-anchor node only weakens the interaction by reducing hyperedge cardinality. (C) Macroscopic organization of connectivity in a directed hypergraph, illustrating the three giant components: the HGSCC, comprising mutually reachable nodes and hyperedges; the HGIN, consisting of elements that can reach the HGSCC; and the HGOUT, containing elements reachable from the HGSCC. (D) Directional message-passing processes underlying percolation. Forward propagation ($w^{(+)}, v^{(+)}$) identifies the HGOUT, backward propagation ($w^{(-)}, v^{(-)}$) identifies the HGIN, and nodes reached by both processes form the HGSCC.
• Figure 2: Percolation in directed hypergraphs with increasing anchor-node fraction. Relative sizes of the HGOUT, HGIN, and HGSCC are shown as functions of the node survival probability $p_N$ for synthetic directed hypergraphs with anchor-node probability $\theta$. Symbols denote Monte Carlo simulations, while solid lines indicate analytical predictions from the message-passing theory. Increasing $\theta$ systematically shifts the percolation threshold toward larger $p_N$, indicating enhanced structural fragility and reduced robustness. Panels correspond to $\theta = 0$ (A), $0.4$ (B), $0.8$ (C), and $1$ (D). All results are obtained for random directed hypergraphs with $N = 10^4$ nodes and $M = 10^4$ hyperedges, with fixed cardinalities $m^{\mathrm{\rm in}} = 3$ and $m^{\mathrm{\rm out}} = 2$.
• Figure 3: Phase diagrams of directed hypergraph percolation with anchor-node heterogeneity. Equilibrium sizes of the (A) HGOUT ($R^{(+)}$), (B) HGIN ($R^{(-)}$), and (C) HGSCC ($R$) are shown over the parameter space defined by the node survival probability $p_N$ and the anchor-node probability $\theta$. Colors indicate the relative size of the corresponding giant component, ranging from the fragmented phase ($R \approx 0$) to the percolating phase. Dashed lines denote the theoretical percolation thresholds obtained from linear stability analysis. All results are shown for random directed hypergraphs with $N = 10^4$ nodes and $M = 10^4$ hyperedges, with fixed cardinalities $m^{\mathrm{\rm in}} = 3$ and $m^{\mathrm{\rm out}} = 2$. Although all three giant components emerge at the same critical boundary, their post-critical growth is governed by distinct critical exponents, with the HGSCC exhibiting systematically slower scaling.
• Figure 4: Universality map of directed hypergraph percolation. Universality classes of the critical exponent $\beta$ in the parameter space defined by the node degree exponent $\gamma_q$ and the hyperedge cardinality exponent $\gamma_m$. Panels correspond to (A) uncorrelated networks without anchor nodes ($\theta = 0$), (B) uncorrelated networks with anchor nodes ($\theta > 0$), (C) maximally correlated networks without anchor nodes ($\theta = 0$), and (D) maximally correlated networks with anchor nodes ($\theta > 0$). Colored regions delineate distinct universality classes, including mean-field, node-dominated, hyperedge-dominated, intermediate, and anomalous regimes. Analytical expressions for $\beta$ are indicated within each region.
• Figure 5: Finite-size scaling in directed hypergraph percolation.(A) Scaling of the pseudo-critical point, showing the offset $p_c(N) - p_c$ as a function of network size $N$. (B) Scaling of the percolation strength $R(p_c)$, evaluated at the pseudo-critical point, as a function of $N$. Symbols show numerical results for the HGOUT (circles), HGIN (squares), and HGSCC (triangles). Dashed lines indicate power-law fits, demonstrating critical scaling behavior. All results are obtained for random directed hypergraphs with parameters identical to those in Fig. \ref{['fig:r1']}.