  Many real systems are strongly characterized by collective cooperative
phenomena whose existence and properties still need a satisfactory explanation.
Coherently with their collective nature, they call for new and more accurate
descriptions going beyond pairwise models, such as graphs, in which all the
interactions are considered as involving only two individuals at a time.
Hypergraphs respond to this need, providing a mathematical representation of a
system allowing from pairs to larger groups. In this work, through the use of
different hypergraphs, we study how group interactions influence the evolution
of cooperation in a structured population, by analyzing the evolutionary
dynamics of the public goods game. Here we show that, likewise network
reciprocity, group interactions also promote cooperation. More importantly, by
means of an invasion analysis in which the conditions for a strategy to survive
are studied, we show how, in heterogeneously-structured populations,
reciprocity among players is expected to grow with the increasing of the order
of the interactions. This is due to the heterogeneity of connections and,
particularly, to the presence of individuals standing out as hubs in the
population. Our analysis represents a first step towards the study of
evolutionary dynamics through higher-order interactions, and gives insights
into why cooperation in heterogeneous higher-order structures is enhanced.
Lastly, it also gives clues about the co-existence of cooperative and
non-cooperative behaviors related to the structural properties of the
interaction patterns.
