Sequential dual-threshold cascade dynamics on random hypergraphs

Existing cascade models on hypergraphs often assume homogeneous node roles and single-stage synchronous activation, leaving hierarchical sequential decision-making insufficiently represented in settings where leader consensus must precede follower mobilization. To address this issue, we propose a two-stage threshold cascade model on role-differentiated hypergraphs. Once the leader threshold ϕ 1 is met, all remaining leaders become active; follower activation is then evaluated against the overall threshold ϕ 2 . We derive a self-consistent equation for the steady-state cascade size using a scalar mean-field closure for sparse, locally tree-like random hypergraphs and validate the theoretical predictions through numerical simulations on Poisson and fixed-size hypergraphs. Our results show hybrid-type discontinuous transitions with finite jumps and finite-size rounded susceptibility peaks, with the leader threshold ϕ 1 strongly controlling cascade initiation. We further find that the “small leader—large follower” configuration can reduce the transition seed fraction and identify a non-monotonic discrete truncation effect in fixed-size hypergraphs. This work provides a theoretical framework for hierarchical contagion in higher-order networks.