Explosive Cooperation in Social Dilemmas on Higher-Order Networks

Understanding how cooperative behaviors can emerge from competitive interactions is an open problem in biology and social sciences. While interactions are usually modeled as pairwise networks, the units of many real-world systems can also interact in groups of three or more. Here, we introduce a general framework to extend pairwise games to higher-order networks. By studying social dilemmas on hypergraphs with a tunable structure, we find an explosive transition to cooperation triggered by a critical number of higher-order games. The associated bistable regime implies that an initial critical mass of cooperators is also required for the emergence of prosocial behavior. Our results show that higher-order interactions provide a novel explanation for the survival of cooperation.

Figure 1

Higher-order games on a hypergraph. The orange triangular areas are hyperedges of size $q_g=3$, corresponding to games played by three players (3-games), while the purple segments are hyperedges of size $q_g=2$, representing pairwise games (2-games). The payoff structures of symmetric 2-games and 3-games are reported in the two boxes.

Figure 2

(a) Fraction of cooperators at equilibrium for the PD on random hypergraphs with $N=1500$, $⟨k⟩=20$ and tunable ratio $\delta$ of three-player interactions. (b)–(e) Quasistationary distributions for $a=1$ and four values of $\delta$. Continuous curves and symbols represent the simulation results averaged over 1500 runs, while dashed lines are the analytical mean-field predictions of Eqs. (3) and (4). (f) Susceptibility $\chi$ as a function of $\delta$ for increasing $N$. (g) Scaling of the position and height (inset) of the peak of $\chi$. The black line is the fitting.

Figure 3

Basins of attraction and critical mass of cooperators for the PD on random hypergraphs. (a) Temporal evolution of the fraction of cooperators for various initial conditions and $\delta =0.4$, $a=1.5$, $⟨k⟩=20$. (b) Unstable stationary state ${\rho }_{-}^{*}$ as a function of $\delta$ for average hyperdegree $⟨k⟩=20$ and different values of $a$. Symbols show the simulation results, while the lines are the analytical mean-field predictions. The shaded areas represent the errors.