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 “ ℓ -hyperedge approximation,” an analytical framework for stochastic processes on regular hypergraphs, in which each individual belongs to k groups of size ℓ . 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 b / c > k rule for cooperation to the ℓ -player donation game; and it yields critical benefit-to-cost ratios for the nonlinear ℓ -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.