Title: Role-separating ordering in social dilemmas controlled by topological frustration,
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.