Degree mixing in multilayer networks impedes the evolution of cooperation

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.

💡 Research Summary

The paper investigates how degree mixing—i.e., assortative (high‑degree nodes preferentially linking to other high‑degree nodes) or disassortative (high‑degree nodes linking to low‑degree nodes)—affects the evolution of cooperation in multilayer networks. The authors construct a two‑layer system of scale‑free networks, each with an average degree of four. One layer serves as the interaction network where players collect payoffs from a pairwise game (prisoner’s dilemma, snowdrift, or stag‑hunt), while the other layer functions as the updating network where players select a neighbor and possibly adopt its strategy. The two layers can have independent degree‑mixing coefficients, denoted AI for the interaction layer and AU for the updating layer. Positive values of A indicate assortative mixing, negative values indicate disassortative mixing.

The networks are generated using the Barabási–Albert model, then rewired with the Xulvi‑Brunet‑Sokolov algorithm to achieve the desired mixing while preserving the degree distribution. The authors explore all nine combinations of AI and AU in the range