Extended Data Figure 1: Further numerical and simulation results.

a, To assess accuracy of our results for empirically plausible selection strength, we performed Monte Carlo simulations with δ = 0.025 and c = 1. This corresponds to a fitness cost of 2.5%, which was determined to be the cost of cooperative behaviour in yeast⁴³. Markers indicate population size times frequency of fixation for a particular value of b on a particular graph. Dashed lines indicate (b/c)^* as calculated from equation (2). All graphs have size N = 100. Graphs are: Barabási–Albert (BA) with linking number m = 3, small world³⁵ (SW) with initial connection distance d = 3 and edge creation probability p = 0.025, Klemm–Eguiluz³⁹ (KE) with linking number m = 5 and deactivation parameter μ = 0.2, and Holme–Kim³⁸ (HK) with linking number m = 2 and triad formation parameter P = 0.2. b, We computed (b/c)^* for 4 × 10⁴ large random graphs (sizes 300–1,000) using four random graph models: Erdös–Rényi³⁴ (ER) with edge probability 0 < p < 0.25, Klemm–Eguiluz with linking number 3 ≤ m ≤ 5 and deactivation parameter 0 < μ < 0.15, Holme–Kim with linking number 2 ≤ m ≤ 4 and triad formation parameter 0 < P < 0.15, and a meta-network⁴² of shifted-linear preferential attachment networks⁴⁰ (Island Barabási–Albert) with 0 < p_inter < 0.25; see Methods for details. c, d, We computed the structure coefficient²⁷ σ = ((b/c)^* + 1)/((b/c)^* − 1) for the same ensemble of random graphs as in Fig. 4 of the main text. Strategy A is favoured over strategy B under weak selection if σa + b > c + σd; see equation (3) of Methods. c, Plot of σ versus , which is the σ-value for a regular graph of the same mean degree . d, Plot of σ versus , which is the σ-value one would expect if the condition (as described in ref. 26) were exact. Here, is the expected degree of a neighbour of a randomly chosen vertex.