Fig. 1

Evolutionary dynamics on scale-free networks embedded into hyperbolic space. a Illustration of the hyperbolic spatial structure (Poincaré Disc) underlying a synthetic network generated by the model described in Methods. The network shown here has N = 2000 nodes, a power-law exponent γ = 2.6, mean degree \(\left\langle k \right\rangle\) ≈ 6, and mean local clustering coefficient \(\bar c = 0.6\) (temperature \(\bar T = 0.3\) as in Eq. (6), see Methods section for details). Hubs, i.e. high-degree nodes, are placed closer toward the center of the disc (lower radial coordinate). The angular space represents the similarity between nodes, such that nodes tend to connect to other nodes close to them in this space. The green line shows the hyperbolic disc of radius R (Eq. (5)) around the green node. We highlight the neighbors of the green node in red. For a high \(\bar c\) (i.e., low \(\bar T\)), as shown here, the green node is highly likely to connect to other nodes within the disc (green line), and very unlikely to connect to nodes outside of it (the further apart, the less probable). The high mean local clustering coefficient is then a consequence of the triangle inequality in the metric space. b Synthetic network generated with the same model but with mean local clustering coefficient \(\bar c = 0.25\) (temperature \(\bar T = 0.7\)). We again show the hyperbolic disc of radius R (Eq. (5)) around the green node and highlight its neighbors in red. Note that due to the higher temperature more long-range connections are formed, i.e., the green node connects to more nodes outside of the disc as compared to a, and does not connect to some node inside of the disc. This effect reduces the mean local clustering coefficient as it induces randomness in the link formation process. c Illustration of evolutionary game dynamics. In structured populations, individuals play with their neighbors in a network. In each game, they generate a payoff given by the payoff matrix (Eq. (1)). After each round, they choose a random neighbor and imitate her strategy with a probability (Fermi–Dirac distribution) that depends on the difference between their payoffs