# Role-separating ordering in social dilemmas controlled by topological frustration

## TL;DR

This paper investigates how topological frustration in a triangular lattice affects the evolution of cooperation in social dilemmas, revealing complex spatial patterns and a method to measure frustration levels.

## Contribution

It introduces the concept of topological frustration's impact on social dilemma dynamics and demonstrates how lattice topology influences cooperation levels and pattern formation.

## Key findings

- Complex honeycomb-like spatial patterns emerge due to topological frustration.
- Cooperation levels serve as indicators of the degree of topological frustration.
- Bridging triangular and square lattices shows the influence of network topology on social dynamics.

## Abstract

"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 anti-coordination 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.