  Traditionally, the evolution of cooperation has been studied on single,
isolated networks. Yet a player, especially in human societies, will typically
be a member of many different networks, and those networks will play a
different role in the evolutionary process. Multilayer networks are therefore
rapidly gaining on popularity as the more apt description of a networked
society. With this motivation, we here consider 2-layer scale-free networks
with all possible combinations of degree mixing, wherein one network layer is
used for the accumulation of payoffs and the other is used for strategy
updating. We find that breaking the symmetry through assortative mixing in one
layer and/or disassortative mixing in the other layer, as well as preserving
the symmetry by means of assortative mixing in both layers, impedes the
evolution of cooperation. We use degree-dependent distributions of strategies
and cluster-size analysis to explain these results, which highlight the
importance of hubs and the preservation of symmetry between multilayer networks
for the successful resolution of social dilemmas.
