In network science, change is good: Historically, network science focused on static networks, in which nodes are connected by permanent links. However, in networked systems ranging from protein-protein interactions to social networks, links change. Although it might seem that permanent links would make it easier to control a system, Li et al. demonstrate that temporality has advantages in real and simulated networks. Temporal networks can be controlled more efficiently and require less energy than their static counterparts. Science , this issue p. 1042

Cooperation is key to the survival of all biological systems. The spatial structure of a system constrains who interacts with whom (interaction partner) and who acquires new traits from whom (role model). Understanding when and to what degree a spatial structure affects the evolution of cooperation is an important and challenging topic. Here, we provide an analytical formula to predict when natural selection favours cooperation where the effects of a spatial structure are described by a single parameter. We find that a spatial structure promotes cooperation (spatial reciprocity) when interaction partners overlap role models. When they do not, spatial structure inhibits cooperation even without cooperation dilemmas. Furthermore, a spatial structure in which individuals interact with their role models more often shows stronger reciprocity. Thus, imitating individuals with frequent interactions facilitates cooperation. Our findings are applicable to both pairwise and group interactions and show that strong social ties might hinder, while asymmetric spatial structures for interaction and trait dispersal could promote cooperation.

The evolution of populations is influenced by many factors, and the simple classical models have been developed in a number of important ways. Both population structure and multiplayer interactions have been shown to significantly affect the evolution of important properties, such as the level of cooperation or of aggressive behavior. Here we combine these two key factors and develop the evolutionary dynamics of general group interactions in structured populations represented by regular graphs. The traditional linear and threshold public goods games are adopted as models to address the dynamics. We show that for linear group interactions, population structure can favor the evolution of cooperation compared to the well-mixed case, and we see that the more neighbors there are, the harder it is for cooperators to persist in structured populations. We further show that threshold group interactions could lead to the emergence of cooperation even in well-mixed populations. Here population structure sometimes inhibits cooperation for the threshold public goods game, where depending on the benefit to cost ratio, the outcomes are bistability or a monomorphic population of defectors or cooperators. Our results suggest, counterintuitively, that structured populations are not always beneficial for the evolution of cooperation for nonlinear group interactions.

What drives the stability, or instability, of complex ecosystems? This question sits at the heart of community ecology and has motivated a large body of theoretical work exploring how community properties shape ecosystem dynamics. However, the overwhelming majority of current theory assumes that species interactions are instantaneous, meaning that changes in the abundance of one species will lead to immediate changes in the abundances of its partners. In practice, time delays in how species respond to one another are widespread across ecological contexts, yet the impact of these delays on ecosystems remains unclear. Here we derive a new body of theory to comprehensively study the impact of time delays on ecological stability. We find that time delays are important for ecosystem stability. Large delays are typically destabilizing but, surprisingly, short delays can substantially increase community stability. Moreover, in stark contrast to delay-free systems, delays dictate that communities with more abundant species can be less stable than ones with less abundant species. Finally, we show that delays fundamentally shift how species interactions impact ecosystem stability, with communities of mixed interaction types becoming the most stable class of ecosystem. Our work demonstrates that time delays can be critical for the stability of complex ecosystems.

Cooperation is ubiquitous ranging from multicellular organisms to human societies. Population structures indicating individuals' limited interaction ranges are crucial to understand this issue. But it remains unknown to what extend multiple interactions involving nonlinearity in payoff influence the cooperation in structured populations. Here we show a rule, which determines the emergence and stabilization of cooperation, under multiple discounted, linear, and synergistic interactions. The rule is validated by simulations in homogenous and heterogenous structured populations. We find that the more neighbours there are the harder for cooperation to evolve for multiple interactions with linearity and discounting. For synergistic scenario, however, distinct from its pairwise counterpart, moderate number of neighbours can be the worst, indicating that synergistic interactions work with strangers but not with neighbours. Our results suggest that the combination of different factors which promotes cooperation alone can be worse than that with every single factor.