Selection plays an important role in evolutionary process, which determines the direction of evolution. The absence of selective difference is called neutral selection (β = 0). When selection acts, the transition probability becomes payoff dependent, which can be further classified into constant and frequency dependent selections. Generally, the probability of one strategy replaces another is fairly complicated under frequency dependent selection55. Fortunately, many results have been obtained with the assumption of weak selection, where perturbation analysis analytically facilitates the derivation56,57. For non-weak selection, the approach of perturbation analysis does not work and the result under weak selection can be violated58.

Neutral selection: β = 0

When β = 0, the selection dynamics are neutral, in which the evolutionary direction is random and independent of the players' payoffs. In this case, Eq. (4) is simplified as

Note that the points in line x 2,A = x 1,A are the equilibria of Eq. (5). The corresponding eigenvalues of the Jacobian matrix at x 2,A = x 1,A are λ 1 = 0 and λ 2 = −2. Thus, the line x 2,A = x 1,A is always a stable manifold and system (5) always reaches consensus. Furthermore, all the consensus states are stable, but none are evolutionarily stable, i.e., final consensus can go back and forth from one state to another (See Fig. 2).

Figure 2 System (5) reaches consensus under neutral selection (β = 0), with N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9. (a) Phase portrait of Eq. (5) where the direction of the velocity field is indicated by the arrows. The ultimate values of the frequency of strategy A in Group-1 and Group-2 converge to the line x 2,A = x 1,A . (b) Simultaneous time-evolution of frequency of strategy A in Group-1 and Group-2. One of the consensus states is shown, where the initial frequency of strategy A in Group-1 and Group-2 is selected randomly. Full size image

Weak selection: β ≪ 1

When β ≪ 1, the effects of payoff differences are small, such that the evolutionary dynamics mainly results from random fluctuations. This case of weak selection means one phenotype is slightly advantageous over another59. In evolutionary biology and population genetics, it is widely accepted that most mutations confer small selective difference60,61.

For simplicity, we assume N 1 = N 2 . We verify the general case of N 1 ≠ N 2 in numerical simulations (See Figs. 3,4,5,6,7). When N 1 = N 2 , Eq. (4) becomes

where is constant, , , , and . Note that κ only influences the time-scale with none effect on the long run evolutionary outcome.

Figure 3 Phase portrait of Eq. (6) under weak selection (β = 0.01). The direction of the velocity field is indicated by arrows. We set N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9, ω = 0.01. (a) When and (a = 2, b = 1, c = 1.5, d = 0, k AA = 0.4, k AB = 0.6 and k BB = 0.8), system (6) reaches an asymptotically stable consensus state of all-A. (b) When and (a = 1.5, b = 0, c = 2, d = 1, k AA = 0.6, k AB = 0.4 and k BB = 0.8), system (6) reaches an asymptotically stable consensus state of all-B. Full size image

Figure 4 Phase portraits of Eq. (6) under weak selection (β = 0.01). The direction of the velocity field is indicated by arrows. We set N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9, ω = 0.01. When and (a = 1.5, b = 1, c = 3, d = 0, k AA = 0.6, k AB = 0.4 and k BB = 0.8), we show two cases of phase portraits when the intra-group bias α changes. (a) For α < α* = 0.65 (α = 0.3), the velocity field diverges to the corners near (0,1) and (1,0), which indicates that system (6) cannot reach an asymptotically stable consensus state. (b) For α > α* = 0.65 (α = 0.7), the velocity field converges to the interior equilibrium, which means that system (6) reaches an asymptotically stable consensus at the interior equilibrium. Full size image

Figure 5 Simultaneous time-evolution of the frequency of strategy A in Group-1 and Group-2 under weak selection (β = 0.01). Initially, strategy A is randomly distributed in Group-1 and Group-2. We set N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9, ω = 0.01, a = 1.5, b = 1, c = 3, d = 0, k AA = 0.6, k AB = 0.4 and k BB = 0.8. Full size image

Figure 6 Phase portrait of Eq. (6) under weak selection (β = 0.01). The direction of the velocity field is indicated by arrows. We set N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9, ω = 0.01. When , and α > α* = 0.65 (a = 2, b = 0, c = 1.5, d = 1, k AA = 0.4, k AB = 0.6, k BB = 0.8 and α = 0.7), the velocity field converges to the corner equilibrium E 1 or E 2 , which means system (6) reaches an asymptotically stable consensus relying on the initialization. Full size image

Figure 7 Simultaneous time-evolution of the frequency of strategy A in Group-1 and Group-2 under weak selection (β = 0.01). We set N 1 = 150, N 2 = 100, L 1 = 6, L 2 = 9, ω = 0.01, a = 2, b = 0, c = 1.5, d = 1, k AA = 0.4, k AB = 0.6, k BB = 0.8. (a) When and , system (6) reaches an asymptotically stable consensus state of all-A. (b) When but , system (6) reaches an asymptotically stable consensus state of all-B. Full size image

From Eq. (6), we get three possible equilibria, i.e., E 1 = (0, 0), E 2 = (1, 1), . E 1 and E 2 are at the corners, which denote that system (6) is composed of all-B and all-A, respectively. E 3 is an interior equilibrium, which means that system (6) consists of A and B players (See Section 4 in SI for details about the stability of each equilibrium). With different parameters c i (1 ≤ i ≤ 5), we further discuss the consensus for system (6) under weak selection.

Case 1: Consensus of the dominant-type game.

When and , strategy A dominates strategy B in the interdependent populations, if k AA < k AB < k BB . This result indicates that all players are more likely to interact with strategy A. In this case, E 1 is an unstable equilibrium and E 2 is a stable equilibrium, which implies that the set of all-A (E 2 ) is the unique evolutionary stable state of this game. Let us take the prisoner's dilemma (PD) game as an example, in which strategies A and B denote defection and cooperation, respectively. The frequency of strategy A in each group converges to the state of all-A and system (6) reaches an asymptotically stable consensus state (See Fig. 3(a)).

When and , strategy B dominates strategy A in the interdependent populations, if k AA > k AB > k BB . Therefore, all players are more likely to interact with strategy B. In this case, E 1 is a stable equilibrium and E 2 is an unstable equilibrium. This situation is similar to that of the above discussion, which leads to the dynamics ending in the state of all-B (E 1 ) and system (6) reaches an asymptotically stable consensus state, as shown in Fig. 3(b).

Therefore, when the interactions between players under the payoff matrix M′ are of the dominant-type game, two interactive groups reach a homogeneous consensus, i.e., the stable consensus state consists of all-A, or all-B.

Case 2: Consensus of the coexisting-type game.

When , and , both E 1 and E 2 are unstable and E 3 is a saddle point, if k AA > k AB and k BB > k AB . This implies that all players are inclined to interact with the opponents of opposed strategies as themselves'. In this case, the velocity field is out of order and system (6) cannot reach a consensus state (See Fig. 4(a)). However, when the intra-group attaching bias α exceeds the critical value α*, E 3 becomes stable and strategies A and B coexist in this interior stable equilibrium. The final state of the system converges to the asymptotically stable consensus state E 3 (See Fig. 4(b)). The corresponding representative model is the snowdrift game. Therefore, when interactions between players with the payoff matrix M′ are of the coexisting-type game and the intra-group attaching bias exceeds the critical value α*, two interactive groups converge to a non-homogeneous consensus, i.e., the stable consensus state coexists with A and B.

For populations with two static topology structures, the final state converges to the interior equilibrium ( , )33. For dynamic interdependent populations, the final dynamics ends at an interior equilibrium E 3 which results from the co-evolution of strategy and structure. In addition, although the intra-group attaching bias α does not affect the interior equilibrium E 3 , it determines the stability of interior equilibrium. Only when α > α*, the interior equilibrium is stable, i.e., a too strong rewiring propensity between two interactive groups (small α) does not benefit the formation of an asymptotically stable consensus. To illustrate the effects of the intra-group attaching bias between two groups on consensus, we present the frequency of strategy A in both groups as shown in Fig. 5. We observe that for α < α*, system (6) cannot reach a consensus state (See Fig. 5(a) and Fig. 5(b)). When increasing the value of α until α > α*, system (6) reaches an asymptotically stable consensus at interior equilibrium E 3 . Interestingly, the increasing of α not only facilitates the formation of a consensus state in an interior equilibrium, but also enhances the speed of reaching consensus (See Fig. 5(c) and (d)). Note that the eigenvalues of the Jacobian matrix at E 3 are and , respectively. Thus, increasing α means decreasing the eigenvalues of λ 2 , which indicates that system (6) reaches the asymptotically stable consensus state at a higher speed.

Case 3: Consensus of the bistable-type game.

When , and α > α*, both E 1 and E 2 are stable and E 3 is unstable. Since the final state converges to E 1 or E 2 , system (6) in the whole population is all-A or all-B, if k AA < k AB and k BB < k AB . This result indicates that all players are inclined to interact with the opponents of same strategies as themselves'.

Besides, when α < α*, E 3 becomes a saddle-point, so the intra-group attaching bias α does not affect global stability of system (6). A representative model is the coordination game. In this case, strategies A and B are bistable in the interdependent populations and system (6) reaches an asymptotically stable consensus state at E 1 or E 2 (See Fig. 6). Note that the equilibrium, to which the system converges, depends on the initial fraction of A and the interior unstable equilibrium ( , ), where . If the initial condition , system (6) converges to all-A; otherwise, to all-B. The effects of initialization on the frequency of strategy A in both groups are shown in Fig. 7. Therefore, when the interactions between players under the payoff matrix M′ are of the bistable-type game, two interactive groups converge to a homogeneous consensus state of all-A, or all-B, which relies on the initialization.

Non-weak selection

For non-weak selection intensity, the perturbation analysis does not work. Besides, the results derived under weak selection fail to extend to strong selection intensity58,62,63. Therefore, we numerically simulate to illustrate the consensus state in interdependent populations. It is shown that: when the consensus state refers to the homogeneous population with only one strategy, the intensity of selection does not change the final convergence state (See Fig. 8(a), (b) and (d)); when the consensus state refers to the non-homogeneous population with coexistent strategy, it will loses its stability under non-weak selection, which drives the two populations to non-consensus states (See Fig. 8(c)). Intuitively, the homogeneous consensus is robust for perturbations, so that very strong selection can not force the consensus being extinct. While the non-homogeneous consensus states is relatively unstable and sufficiently strong selection can force the consensus deviating from the previous consensus state. Therefore, the homogeneous consensus states in interdependent populations are robust in arbitrary selection intensity.