Although the cooperative dynamics emerging from a network of interacting players has been exhaustively investigated, it is not yet fully understood when and how network reciprocity drives cooperation transitions. In this work, we investigate the critical behavior of evolutionary social dilemmas on structured populations by using the framework of master equations and Monte Carlo simulations. The developed theory describes the existence of absorbing, quasi-absorbing, and mixed strategy states and the transition nature, continuous or discontinuous, between the states as the parameters of the system change. In particular, when the decision-making process is deterministic, in the limit of zero effective temperature of the Fermi function, we find that the copying probabilities are discontinuous functions of the system's parameters and of the network degrees sequence. This may induce abrupt changes in the final state for any system size, in excellent agreement with the Monte Carlo simulation results. Our analysis also reveals the existence of continuous and discontinuous phase transitions for large systems as the temperature increases, which is explained in the mean-field approximation. Interestingly, for some game parameters, we find optimal "social temperatures" maximizing/minimizing the cooperation frequency/density.

Cultured neuronal networks (CNNs) have recently achieved major relevance as an alternative to in vivo models. While many works investigate the evolution of functional connectivity alone, experimental evidence of the simultaneous change of the structural neuronal network substrate is scarce. In the present study, we monitored the coevolution of structural and functional connectivities of neuronal cultures grown on top of microelectrode arrays, in a setup that allows the simultaneous recording of electrophysiological signals and microphotography detailed to the single-link level. During the observed 3 weeks lifespan, initially isolated invertebrate neurons form an ex novo complex circuitry of neuronal aggregates characterized by a small-world topology, with abundant neuronal loops (high clustering) and short distances. At the same time, the observation of synchronization events among electrodes reaches a maximum, coinciding with the spatial percolation of the neuronal network. At this stage, the correlation between the structural and the corresponding functional network is the largest, with around 15% of the physical links relating electrodes that fire synchronously. As the culture matures, this correlation smoothly declines but becomes more significant. Finally, at the local level, we found that the electrodes supporting the most coherent activity, the functional hubs, are not hubs in the physical circuitry but nodes with an average degree. This study demonstrates that functional networks of self-organized neuronal systems are discreet proxies of the underlying structure. It paves the way for future investigations to elucidate the intricate relationship between structure and function in other scenarios mediated by external stimulation.