Hyperedge approximation for stochastic processes on higher-order networks.

Graphs provide a natural framework to describe processes shaped by pairwise interactions among agents. But many dynamical systems involve interactions within groups of three or more agents. Here, we develop the "[Formula: see text]-hyperedge approximation," an analytical framework for stochastic processes on regular hypergraphs, in which each individual belongs to [Formula: see text] groups of size [Formula: see text]. The framework accommodates both higher-order interactions that determine payoffs and higher-order processes for updating states in response to payoffs. For evolutionary games on hypergraphs, our analysis generalizes the classical [Formula: see text] rule for cooperation to the [Formula: see text]-player donation game; and it yields critical benefit-to-cost ratios for the nonlinear [Formula: see text]-player public goods game, which remains bounded as the degree grows. Applied to neutral complex contagions, where inheritance of states occurs within hyperedges rather than along parent-offspring edges, the framework gives a closed-form fixation probability, showing how a single complexity parameter governs the spread of rare types. Coupling the two processes produces a unified stochastic model of payoff-biased complex contagions in structured populations. Together, these results extend pair approximation from graphs to hypergraphs, accommodating multiway interactions and group-level inheritance with no pairwise analog.