Hyperedge approximation for stochastic processes on higher-order networks

Graphs are a standard framework for describing dynamical processes shaped by pairwise interactions among agents. But many systems involve interactions in groups of three or more agents. Here, we develop a method of "$\ell$-hyperedge approximation", a framework to analyze stochastic population processes on regular hypergraphs, in which each individual belongs to $k$ groups of size $\ell$. The framework accommodates both higher-order interactions that determine payoffs and higher-order processes for updating states in response to payoffs. Applied to evolutionary game dynamics, the framework generalizes the classical pairwise result on benefits and costs, $b/c>k$, that favors the spread of cooperation; and it provides critical benefit-to-cost ratios for nonlinear $\ell$-player public goods games that cannot be reduced to pairwise interactions. Applied to complex contagions, where inheritance of states occurs within hyperedges rather than along parent-offspring edges, the framework gives a closed-form result for the fixation probability, which shows how a complexity parameter governs the spread of rare types. Coupling the two processes produces a single stochastic model of payoff-biased complex contagion in structured populations. These results extend pair approximation from graphs to hypergraphs, accommodating multi-way interactions and inheritance structures with no pairwise analog.