Role-separating ordering in social dilemmas controlled by topological frustration

‘‘Three is a crowd” is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention. With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game. By using different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary interaction network.

Figure 1

The minimal frustration configuration in the triangular lattice. Every blue (dark gray) site (cooperator) receives the best payoff, although defectors, marked red (light gray), have only half of their connections leading to the best payoff. Frustrated bonds are drawn by dashed lines and bonds that maximize the payoff are drawn by full lines.

Figure 2

The fraction of cooperators in equilibrium as a function of $T$. The results refer to Monte Carlo simulations (symbols) and the master equation ODE (lines) in the logit model for both lattices. Note that simulation and analytical results agree well and reproduce the main characteristic of the system, namely a nonvanishing cooperation level.

Figure 3

The fraction of cooperators in equilibrium as a function of $T$, as obtained by means of Monte Carlo simulations (for $S=0$). Note that the difference between the logit and the imitation model is significant for $T>1$, where the logit model exhibits a minimal cooperation level. Also note that the sharp drop in cooperation occurs in the same region.

Figure 4

Heat maps encoding the cooperation level for the whole $T-S$ plane. The top row, (a) and (b), shows results obtained on the square lattice, while the bottom row, (c) and (d), shows results obtained on the triangular lattice. The left column, (a) and (c), shows results obtained with the imitation dynamics, and the right column, (b) and (d), shows the results obtained with the logit rule. When imitation dynamics is used, there is little difference inferable that would be due to the differences in the interaction lattice. For the logit model, the level of cooperation is higher in the SD region for both topologies. Most interestingly, for the triangular lattice, we can observe two different phases that are separated by a straight line.

Figure 5

The fraction of cooperators in equilibrium along the line defined by $T=2-S$ in the logit model on a triangular lattice. Here the payoff values are varied via the control parameter $r$, where $T=1+r$ and $S=1-r$. Instead of a homogeneous state, like on the square lattice, we observe two different phases with honeycomb-like spatial patterns. The insets illustrate the typical honeycomb cell that is characteristic of each phase.

Figure 6

The level of frustration, $\varphi$, in the snowdrift region of the $T-S$ diagram for the imitation (a) and logit (b) strategy updating rules on the triangular lattice. The imitation model has many frustrated links, around $60\%$, while the logit model maintains the low and homogeneous frustration of around $35\%$.

Figure 7

Typical snapshots of the square lattice in the top row, (a) and (b), and the triangular lattice in the bottom row, (c) and (d), in the SD region. The left column, (a) and (c), shows the results for imitation model while the right column, (b) and (d), shows the results for the logit model. In the logit model on the square lattice a checkerboard pattern quickly emerges. In the logit model on the triangular lattice, on the other hand, we see the honeycomb pattern. Here we use $T=1.2$ and $S=0.5$.

Figure 8

Leading microscopic processes that guide pattern formation in a frustrated topology. (a) A cooperator surrounded by defectors is very stable, since the change in payoff here would be $-6S$. (b) Defectors in the vertices, now shown in the center, may change their strategy, depending on the parameters. The payoff difference would be $3(S-T+1)$. If $S>T-1$ the chance that defector becomes a cooperator is high. $S=T-1$ is the line dividing the two phases seen in Fig. 4.

Figure 9

The effect of link deletion on the cooperation level along the $T=2-S$ line in the logit model. Here the payoff values are varied via the control parameter $r$, where $T=1+r$, while $S=1-r$. The topology is modified gradually where a fraction $X$ of diagonal links are removed from a triangular lattice. Accordingly, $X=0$ corresponds to the triangular lattice and $X=1$ to the square lattice topology. If the topological frustration is mitigated by deleting just a few links, then the steep transition between ordered phases vanishes. Alternatively, when we add a tiny frustration to the interaction graph by leaving $X=1$ then the antiferromagnetic order disappears immediately.